==== TABLE OF CONTENTS ====
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//paper.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//preamble.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/01-introduction.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/02-preliminaries.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/03-graph-calculus.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/04-area-transfer.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/05-maximal-reduction.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/06-stress-extension.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/07-compact-construction.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/08-transmission-analysis.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/09-scalar-error.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/10-completion.tex
The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//references/references.bib
===========================

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//paper.tex ====
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\documentclass[11pt]{article}
\input{preamble}
\title{The Penrose inequality for maximal\\asymptotically hyperbolic initial data}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the sharp Penrose inequality for three-dimensional maximal
asymptotically hyperbolic initial data with spherical conformal infinity.
Under the dominant energy condition, the stated decay and integrability
assumptions, and a future-timelike mass covector, the invariant mass is bounded below
by the Schwarzschild--anti-de Sitter mass associated with the minimum
enclosing area of a weakly future outer-trapped boundary.
The boundary may be disconnected, and no restriction is imposed on the
compact topology.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-preliminaries}
\input{sections/03-graph-calculus}
\input{sections/04-area-transfer}
\input{sections/05-maximal-reduction}
\input{sections/06-stress-extension}
\input{sections/07-compact-construction}
\input{sections/08-transmission-analysis}
\input{sections/09-scalar-error}
\input{sections/10-completion}
\bibliographystyle{plainnat}
\bibliography{references/references}
\end{document}
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//paper.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//preamble.tex ====
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\newcommand{\N}{\mathbb N}
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==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//preamble.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/01-introduction.tex ====
\section{Introduction and statement of the result}
\label{sec:hypotheses}

The Penrose inequality relates the mass of an initial-data set to the
area needed to enclose a trapped region. Penrose's original argument
\citep{Penrose1973} connects such a bound with gravitational collapse
and the expected behavior of black holes. In the asymptotically flat
Riemannian case, Huisken and Ilmanen proved the connected-horizon
inequality by weak inverse mean curvature flow \citep{HI2001}, and
Bray proved the inequality for the total area of a possibly disconnected
horizon \citep{Bray2001}. Bray and Lee extended the Riemannian result to
dimensions below eight \citep{BrayLee2009}.

For negative cosmological constant the natural comparison metric is
Schwarzschild--anti-de Sitter. Its horizon of area $4\pi a^2$ has mass
$(a+a^3)/2$ when the cosmological constant is $-3$.
The additional cubic term is accompanied by a different asymptotic mass:
the four fluxes at spherical conformal infinity form a Lorentz covector
\citep{CH2003}. For asymptotically hyperbolic graphs,
Dahl, Gicquaud, and Sakovich established mass identities and Penrose-type
bounds \citep{DGS2013}; de Lima and Gir\~ao proved the sharp bound for
a class of balanced graphs whose horizons are star-shaped and mean
convex in the reference slice
\citep{deLimaGirao2016}. Directly passing to infinity along
inverse mean curvature flow has a distinct obstruction in this setting
\citep{Neves2010}.

Here we prove the global maximal asymptotically hyperbolic Penrose
inequality in the class specified below. Thus the corresponding maximal
formulation of the Penrose conjecture is resolved positively, with
arbitrary compact topology and without an outermostness assumption on
the original boundary. The area in the statement is the minimum
\emph{enclosing} area, not necessarily the area of that boundary.
The proof does not require a spacetime development or an additional
null-expansion condition. We make no assertion about nonmaximal
initial data or equality rigidity.

\subsection{Geometry, decay, and mass}

Let $(\Omega,g)$ be a smooth connected orientable three-manifold with
nonempty compact smooth boundary $S$. The metric space obtained by
including $S$ is complete. There is one end, with coordinates
$(r,\omega)\in(r_0,\infty)\times\Sph^2$ outside a compact region.
Write
\begin{equation}\label{eq:background}
 b=\frac{\dd r^2}{1+r^2}+r^2\sigma,
 \qquad \eta=\operatorname{arsinh}r,
 \qquad V_0=\sqrt{1+r^2},\qquad V_i=r\omega_i\quad(1\le i\le3),
\end{equation}
where $\sigma$ is the unit round metric and $\omega_i$ are the coordinate
functions of the unit sphere. The Laplacian is $\Delta=\divg\nabla$;
mean curvature is the sum of the principal curvatures.

Fix $0<\alpha<1$ and $3/2<\tau<3$. On the end, a tensor belongs to
$C^{k,\alpha}_\tau$ if its $b$-covariant derivatives through order $k$
are $O(e^{-\tau\eta})$, and its order-$k$ local H\"older seminorms
on uniform $b$-unit coordinate balls have the same weight. Equivalently,
the $C^{k,\alpha}$ norms of its components in uniform background charts,
multiplied by $e^{\tau\eta}$ at the chart centers, are bounded.
This convention is a tensor-norm convention. It does not mean
Euclidean coordinate-component decay. On compact subsets, including
the finite boundary, all fields below are smooth.

Let $K$ be a smooth symmetric covariant two-tensor on $\Omega$, smooth
up to $S$, and assume
\begin{equation}\label{eq:decay}
 g-b\in C^{2,\alpha}_\tau,
 \qquad K\in C^{1,\alpha}_\tau.
\end{equation}
For $e=g-b$, define the mass fluxes in the chosen chart by
\begin{align}\label{eq:ah-mass}
 p_\mu(g)&=\frac1{16\pi}\lim_{R\to\infty}
       \int_{\{r=R\}}\mathbb U(V_\mu,e)(\nu_b)\dd A_b,
       &&0\le\mu\le3,\\
 \mathbb U(V,e)&=V(\divg_b e-\dd\tr_b e)
              +(\tr_b e)\dd V-e(\nabla_b V,\cdot).
              \label{eq:mass-integrand}
\end{align}
Here $(\divg_b e)_i=\nabla_b^j e_{ji}$, and $\nu_b$ points toward
increasing $r$. Every operator and measure in this definition uses $b$.
We require the four limits to exist and be finite. When the covector
is future timelike, its invariant mass is
\begin{equation}\label{eq:invariant-mass}
 m_{\AH}(g)=\sqrt{p_0(g)^2-p_1(g)^2-p_2(g)^2-p_3(g)^2},
 \qquad p_0(g)>\sqrt{p_1(g)^2+p_2(g)^2+p_3(g)^2}.
\end{equation}
The spatial entries in this covector are metric mass charges; they are
not ADM momentum or angular momentum.

\subsection{Constraints, boundary, and enclosing area}

We use units $G=c=1$ and cosmological constant $\Lambda=-3$.
Maximality and the noncosmological constraint densities are
\begin{equation}\label{eq:constraints}
 \tr_gK=0,
 \qquad 16\pi\mu=R_g+6-|K|_g^2,
 \qquad 8\pi J_i=\nabla_g^jK_{ij}.
\end{equation}
Assume the dominant energy condition and weighted integrability
\begin{equation}\label{eq:dec-integrability}
 \mu\ge|J|_g,
 \qquad \int_{\mathrm{end}}V_0
       \bigl(\mu+|J|_g+|K|_g^2\bigr)\dd V_g<\infty.
\end{equation}
In particular $R_g+6\ge0$ and its $V_0$-weighted integral is finite.

On each component of $S$, choose the unit normal $\nu$ pointing into
$\Omega$, toward the designated exterior. Set
\begin{equation}\label{eq:future-trapping}
 H=\divg_S\nu,
 \qquad \tr_SK=(g^{ij}-\nu^i\nu^j)K_{ij},
 \qquad \theta_+=H+\tr_SK.
\end{equation}
We assume only $\theta_+\le0$ on every component, with this same future
orientation. For comparison with a spacetime convention, the sign is
$K(X,Y)=\overline g(\overline\nabla_Xn_{\mathrm{future}},Y)$.
There is no assumption on the inward null expansion.

\begin{definition}[Minimum enclosing area]\label{def:minimum-area}
An admissible end-side domain is a connected smooth codimension-zero
submanifold-with-boundary $D\subset\Omega$, closed as a subset of
$\Omega$, such that
\[
 \operatorname{int}D\subset\operatorname{int}\Omega
 \quad\hbox{and}\quad
 \{r>R\}\subset D\quad\hbox{for some }R.
\]
Its entire intrinsic manifold boundary $\partial D$ must be compact,
smooth, embedded, and two-sided, with any portions coinciding with $S$
included. Define
\begin{equation}\label{eq:minimum-area}
 A_{\min}(S;g)=\inf_D\Area_g(\partial D).
\end{equation}
In particular $D=\Omega$ is admissible and contributes $\partial D=S$.
An empty relative frontier is not an admissible replacement for this
intrinsic boundary.
\end{definition}

Neither the number nor the genus of the boundary components is
restricted. No trapping condition is imposed on the competitors in
Definition~\ref{def:minimum-area}.

\begin{theorem}[Global maximal AH Penrose inequality]\label{thm:main}
Under \eqref{eq:decay}--\eqref{eq:future-trapping}, the completeness,
one-end, and smoothness assumptions above, and the future-timelike
condition \eqref{eq:invariant-mass}, one has
\begin{equation}\label{eq:main-inequality}
 m_{\AH}(g)\ge\frac12\bigl(r_A+r_A^3\bigr),
 \qquad r_A=\sqrt{\frac{A_{\min}(S;g)}{4\pi}}.
\end{equation}
The coefficient is sharp: equality holds for every positive-mass
time-symmetric Schwarzschild--anti-de Sitter exterior bounded by its
minimal horizon.
\end{theorem}

The proof establishes a time-component statement for Riemannian data
which is useful independently and which does not require a causal
condition on their mass covector.

\begin{theorem}[Time-symmetric time-component inequality]\label{thm:ts}
Let $(\Omega,g)$ have the smoothness, topology, completeness, and end
structure above. Suppose $g-b\in C^{2,\alpha}_\tau$, $3/2<\tau<3$,
the four fluxes \eqref{eq:ah-mass} are finite, and
\[
 R_g\ge-6,\qquad H_g(S)\le0,
 \qquad \int_{\mathrm{end}}V_0(R_g+6)\dd V_g<\infty.
\]
In every such fixed end chart,
\begin{equation}\label{eq:ts-inequality}
 p_0(g)\ge\frac12(a+a^3),
 \qquad a=\sqrt{A_{\min}(S;g)/(4\pi)}.
\end{equation}
\end{theorem}

\subsection{Structure and new ingredients}

The first deformation solves a finite-height graph problem together
with a positive lapse and a conformal factor. It changes maximal data
into Riemannian AH data with scalar curvature at least $-6$ and a free
minimal boundary. Its mass cost tends to zero. The conformal factor
can contract area, so pointwise comparison alone is insufficient.
A volume penalty and a finite-time weak-flow argument recover the
enclosing-area comparison. A continuous choice of inner flux, based
on all possible perimeter-minimizer contact sets, removes contact with
the original weakly trapped boundary.

For Theorem~\ref{thm:ts}, a nonlinear electrostatic potential produces a
divergence-free stress. Its algebraic constraint encodes the cubic
term $a^3/2$ in the mass budget of an AF attachment. A second graph
construction on a compact interior joined to that attachment yields
an AF comparison exterior to which the ordinary Riemannian Penrose
inequality applies. We prove the coupled transmission estimates and
continuation needed for this construction. Where a trace cutoff can
spoil the scalar-curvature sign, blowup gives uniformly controlled
metric neighborhoods; additional obstacles then put these regions
strictly inside a free minimizing enclosure.

The area comparison uses weak inverse mean curvature flow only over
a fixed interval of relative areas. Its curvature estimate depends on
a scalar lower bound and fixed topology, and accommodates disconnected
leaves. It therefore has a role different from taking an asymptotic
limit of hyperbolic Hawking mass.

The numerical argument can be seen before entering these constructions.
In the positive-area time-symmetric case, fix a metric after the
preliminary approximation and write
$A_{\min}(S;g)=4\pi a^2$. Let $E_0$ denote the comparison enclosure of
the auxiliary caps alone and $E$ the enclosure that also contains the
regions where scalar curvature may be negative. Given small positive
tolerances $\eps_m,\eps_A$, the attachment and comparison constructions,
followed by area transfer, give
\[
 \begin{aligned}
 m_o&\le p_0(g)-a^3/2+\eps_m,
 &m_{\AF}(h')&\le m_o+\eps_m,\\
 \Area_{h'}(\partial E)&\ge\Area_{h'}(\partial E_0)
                    \ge(1-\eps_A)4\pi a^2.
 \end{aligned}
\]
(Propositions~\ref{prop:af-attachment}, \ref{prop:af-comparison},
and~\ref{prop:area-transfer}).
The AF Penrose inequality applies to the exterior of $E$ and yields
\[
 p_0(g)\ge \frac{a^3}{2}
       +\frac a2\sqrt{1-\eps_A}-2\eps_m.
\]
Thus the attachment supplies the cubic term and the AF inequality the
linear term. The errors tend to zero through successive finite choices;
no convergence of the comparison metrics is required.
Theorem~\ref{thm:ah-reduction} then transfers the time-symmetric bound
to maximal data. The original mass covector is put in a rest chart
before any deformation, so its time component is the invariant mass.

Sections~\ref{sec:graph-calculus} and~\ref{sec:area-transfer} develop
the shared identities and area comparison. Section~\ref{sec:maximal-reduction}
reduces Theorem~\ref{thm:main} to Theorem~\ref{thm:ts}.
Sections~\ref{sec:stress-extension}--\ref{sec:scalar-error} construct
the AF comparison for the latter. Section~\ref{sec:completion}
assembles the inequalities and checks sharpness.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/01-introduction.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/02-preliminaries.tex ====
\section{Preliminaries}
\label{sec:preliminaries}

\subsection{Compact cores and auxiliary caps}

\begin{lemma}\label{lem:compact-core}
The sufficiently large coordinate spheres bound connected compact
truncations of $\Omega$. Each component of $S$ can be capped on its
other side to produce a smooth complete connected one-ended manifold
$X$ without boundary, with the orientation extending that of $\Omega$.
Let $O$ denote the compact cap set, including its
boundary. For competitors containing $O$, perimeter depends only on
the metric on $\Omega$ and its boundary trace.
\end{lemma}
\begin{proof}
The end metric is uniformly comparable to $b$ sufficiently far out.
Radial infinity escapes compact subsets: otherwise a sequence tending
to radial infinity would accumulate in a compact coordinate
neighborhood, while pairwise disjoint background balls of a fixed small
radius about a subsequence would have uniformly positive metric
volumes and lie in a fixed compact neighborhood. This is impossible.
A large coordinate sphere separates the product end from an inner
side. If the latter were noncompact, a smooth compact exhaustion would
yield an end separated from the given product end. One may choose this
exhaustion with finitely many complement components at each stage,
since its smooth compact frontier has finitely many components.
Nested extraction then contradicts the one-end assumption. The inner
side is therefore compact. Paths between its points can be pushed
back across the product end, so the truncations can be taken connected.

Every component of $S$ is a closed orientable surface and bounds a
handlebody. Orient each handlebody and glue by a map reversing the
induced boundary orientations.  The collars then extend the orientation
of $\Omega$ across each seam. Extend the metric smoothly across $S$.
The added region is compact; completeness
at infinity is unchanged. For a set containing $O$ up to a null set,
its characteristic function is constant almost everywhere in the
cap interiors. Its perimeter therefore uses no metric values there.
\end{proof}

We use the ordinary perimeter of a finite-perimeter set in the capped
manifold. In particular it is not a relative perimeter that discards
the original boundary.

\begin{lemma}\label{lem:cut-enclosure}
Every admissible cut from Definition~\ref{def:minimum-area} determines
a precompact finite-perimeter enclosure of $O$ whose perimeter is at
most the area of that cut. Conversely, a precompact enclosure of $O$
with smooth free boundary in $\operatorname{int}\Omega$ and connected
exterior gives an admissible cut with the same area. If this enclosure
is outward minimizing, its boundary realizes the minimum enclosing
area of its own closed exterior.
\end{lemma}
\begin{proof}
For an admissible $D$, complement its manifold interior in $X$.
The result contains $O$ and is precompact because $D$ contains the
whole sufficiently distant end. Local boundary coordinates show that
its perimeter is bounded by the entire intrinsic area of $\partial D$,
including any portions on $S$. This also follows by smoothing corners
and passing to the perimeter lower limit. For the converse, take the
closed exterior of the smooth enclosure. It is connected, has the
required end, and has precisely the stated intrinsic boundary.
Any admissible cut of this new exterior, complemented in $X$, encloses
the original enclosure; outward minimization gives the final claim.
\end{proof}

The compactness, lower semicontinuity, coarea, slicing, reduced-boundary
structure, and union/intersection inequalities for finite-perimeter
sets will be used below. These are local facts and hold in smooth
Riemannian coordinate charts; see \citet[Chapters 12--18]{Maggi2012}.
They also give compactness and lower semicontinuity for a uniformly
convergent family of uniformly positive continuous metrics on a fixed
compact set, by comparison with smooth metrics. A free locally
perimeter-minimizing boundary is smooth in ambient dimension three;
one may use the smooth-ambient regularity theorem in
\citet[Chapter 7, Section 5, Theorem 5.8]{Simon2014}, followed by
elliptic bootstrapping. We invoke this regularity only after excluding
contact with every obstacle and outer truncation.

\subsection{Mass covariance and small metric changes}

\begin{proposition}\label{prop:mass-covariance}
Suppose $g-b\in C^{2,\alpha}_\tau$, $\tau>3/2$, and
$\int V_0|R_g+6|\dd V_g<\infty$ on the end. The fluxes
\eqref{eq:ah-mass} transform as a Lorentz covector under hyperbolic
isometries of the asymptotic chart. Such chart changes preserve the
weighted decay and integrability hypotheses used here. Consequently
a future-timelike original covector admits a chart with
$p_0=m_{\AH}$ and $p_1=p_2=p_3=0$.
\end{proposition}
\begin{proof}
The potentials in \eqref{eq:background} are the ambient coordinate
functions of the future unit hyperboloid. Hyperbolic isometries act on
their span by Lorentz transformations. The flux integrand is tensorial
in $b$, linear in $V$, and natural under these isometries.
It remains to justify replacing the transformed integration surfaces
by the original large coordinate spheres. The static identity
$\nabla_b^2V=Vb$ gives
\[
 \divg_b\mathbb U(V,e)=V\,D R_b(e).
\]
The difference between $D R_b(e)$ and $R_g+6$ is bounded by a constant
times
\[
 |e|_b^2+|\nabla_be|_b^2+|e|_b|\nabla_b^2e|_b.
\]
Its $V_0$-weighted integral over $r>D$ is
$O(D^{3-2\tau})$, since $\dd V_b$ is comparable to
$r\dd r\dd A_\sigma$. Also $|V_i|\le V_0$.
Thus the divergence is absolutely integrable on the end for each
potential. The divergence theorem on regions between the two
families of spheres makes their flux difference tend to zero.
This is the covariance mechanism of
\citet[Proposition 2.2, Theorem 2.3, and Section 3]{CH2003} in the
normalization \eqref{eq:ah-mass}.

A fixed hyperbolic isometry changes distance from the chosen origin
by a bounded amount. The exponential weights and $V_0$ are therefore
comparable before and after the change, and uniform background balls
are carried to uniform background balls. The rest-chart assertion is
the elementary Lorentz normal form of a future-timelike covector.
\end{proof}

The boundary sign and the enclosing-area definition are intrinsic.
We will apply Proposition~\ref{prop:mass-covariance} to the original
data before constructing any comparison metric. No assertion that
intermediate covectors remain timelike is required.

If $(1-\eps)g\le\widetilde g\le(1+\eps)g$ as quadratic forms,
then for every two-surface its area changes by factors between
$1-\eps$ and $1+\eps$. Taking infima over the same admissible domains
gives the same comparison for $A_{\min}$. This observation will be
used when passing through the preliminary conformal approximation
and the later corner smoothing.

\subsection{The AF comparison theorem}

For an AF metric $h$ in a Euclidean end chart $y$, our normalization is
\begin{equation}\label{eq:adm-mass}
 m_{\AF}(h)=\frac1{16\pi}\lim_{R\to\infty}
 \int_{|y|=R}(\partial_jh_{ij}-\partial_ih_{jj})
                  n^i\dd A_{\mathrm{Eucl}}.
\end{equation}

\begin{theorem}[AF Riemannian Penrose inequality]\label{thm:af-penrose}
Let $(Y,h)$ be a complete connected one-ended three-dimensional AF
manifold with smooth compact outer-minimizing minimal boundary.
Assume $R_h\ge0$, $h-\delta=O_2(|y|^{-\beta})$ for some
$\beta>1/2$, and $R_h=O(|y|^{-q})$ for some $q>3$.
If the total boundary area is $A$, then
\begin{equation}\label{eq:af-penrose}
 m_{\AF}(h)\ge\sqrt{A/(16\pi)}.
\end{equation}
Disconnected boundary is allowed.
\end{theorem}

This is the dimension-three case of the published
\citet[Theorem 1.4]{BrayLee2009}; its asymptotic definition is on
page 82 and its boundary convention on page 83.
Only the inequality is used. The smoothness, scalar sign, and
outer-minimization required here will be established for the AF
comparison exterior, rather than assumed for either original AH
metric.

\subsection{Local scalar elliptic estimates}

We specify the standard local analytic inputs, separating them from
the coupled transmission estimates proved in
Section~\ref{sec:transmission-analysis}.

\begin{proposition}[Scalar local estimates]\label{prop:elliptic-inputs}
On fixed coordinate balls in dimension three the following estimates
hold, with constants depending on the indicated ellipticity,
coefficient, and geometry bounds.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Bounded measurable uniformly elliptic divergence matrices admit
local subsolution bounds, weak Harnack estimates for nonnegative
supersolutions, and inhomogeneous Harnack and H\"older estimates for
nonnegative solutions. Bounded divergence drifts multiplying the
unknown, vector forcing in $L^q$, $q>3$, and scalar forcing in $L^p$,
$p>3/2$, are permitted in the corresponding estimates.
\item For Laplace equations with uniformly positive Lipschitz metric
coefficients in interior or flattened smooth boundary charts,
divergence forcing in $L^q$ gives local $W^{1,q}$ estimates.
Scalar forcing in $L^2$ can be included for $3<q<6$.
The same statements hold with local zero Dirichlet data or natural
Neumann data, with the divergence forcing included in the conormal
current. A boundary datum in $L^{p_1}$, $p_1>2$, permits a choice
$3<q<3p_1/2$ sufficiently close to $3$.
\item Smooth-coefficient scalar Dirichlet and conormal problems have
the usual Schauder estimates, including divergence-form
$C^{1,\gamma}$ estimates when the uniformly elliptic coefficients
and vector forcing are $C^\gamma$, with boundary data of the
corresponding regularity. These estimates are used only after their
coefficient hypotheses have been obtained.
\item For a bounded measurable symmetric uniformly elliptic matrix
$a$ on a ball, its positive Dirichlet Green kernel is bounded above
by $C|z-z'|^{-1}$, and below by $c|z-z'|^{-1}$ when the pole and
the nearby evaluation point lie in a fixed smaller ball.
\end{enumerate}
\end{proposition}

The local boundedness, Harnack, and H\"older estimates are the scalar
divergence theory of \citet{Serrin1964} and
\citet[Chapter 8]{GT2001}. The Laplace divergence estimates are the
local Calder\'on--Zygmund estimates, with smooth-boundary
localization; see \citet{Stein1970,GT2001}.
For natural Neumann data, an explicit estimate is given by
\citet[Appendix, proof of Lemma 6.1, (6.2), p.\ 19
of arXiv v3]{ChoiKim2013} on bounded $C^1$ localizations.
Our Lipschitz metric coefficients are VMO. A cutoff retains the
full conormal current and the larger-patch $W^{1,2}$ norm.
In dimension three the scalar and boundary forcing exponents are
$3q/(q+3)$ and $2q/3$, respectively, so approximation permits
$3<q<\min\{6,3p_1/2\}$ for the data in part~(ii).
For the boundary exponent in part~(ii), the trace embedding of
$W^{1,q'}$ into $L^{2q/(2q-3)}$ on a two-dimensional face allows
pairing with $L^{p_1}$ whenever $q<3p_1/2$.
Scalar $L^2$ forcing belongs locally to $W^{-1,q}$ when $q<6$.
Part~(iv) is the local Green comparison of
\citet[Theorem 7.1 and the discussion on pp.\ 66--67]{LSW1963}.

Natural homogeneous flux permits even reflection in a flattened
boundary chart, with the full current reflected. Smooth prescribed
flux errors can be placed in a bounded smooth vector field.
Fermi-coordinate metric matrices reflected at a smooth face are
Lipschitz. Constant Dirichlet data for a scalar height permit odd
reflection after subtraction of that constant. We spell out each
reflection where it is needed; none of these scalar statements is a
general nonlinear transmission theorem.

For the electrostatic equation we also use an exact degenerate
gradient regularity statement. Suppose
\[
 A(z,0)=0,\qquad
 \lambda|\xi|I\le A_\xi(z,\xi)\le C|\xi|I,
 \qquad
 |A(z,\xi)-A(z',\xi)|\le C|z-z'|\,|\xi|^2,
\]
with symmetric $A_\xi$, continuous dependence, and smooth dependence
on $\xi\ne0$. Then a bounded weak $W^{1,3}$ solution of
$\partial_iA^i(z,\dd u)=0$ is locally $C^{1,\gamma}$ for some
$\gamma>0$, with uniform estimates from the structural constants
and a local energy bound. Indeed $A_\xi(z,0)=0$ extends continuously,
and integration along a radial segment gives $|A(z,\xi)|\le C|\xi|^2$.
The assumptions of \citet[Theorem 1.1, (1.5)--(1.8)]{AraujoZhang2020}
therefore hold with $n=p=3$, zero forcing in $L^\infty$, and spatial
H\"older exponent $\sigma=1/2$.  We use only an exponent
$0<\gamma<\min\{\alpha_m,1/4\}$, where $\alpha_m>0$ is a
constant-coefficient gradient exponent for the same structural constants.
No optimal exponent or endpoint estimate is required.
If these inequalities initially hold only on a known
bounded gradient range, the constitutive law must first be extended
with the same structure; that extension is constructed in
Section~\ref{sec:stress-extension}.

All global invertibility, weighted end estimates, seam regularity,
continuation, and uniform large-parameter conclusions used later are
proved below. The local estimates in this subsection supply only
their explicitly stated scalar ingredients.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/02-preliminaries.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/03-graph-calculus.tex ====
\section{Static graph identities and differential estimates}
\label{sec:graph-calculus}

This Section proves the differential identities used in both constructions.
They are identities on a Riemannian base; no spacetime satisfying field
equations is assumed.  In particular, the reference tensor in this Section
may be either the given maximal tensor or the synthetic tensor constructed
later.

The curvature, current, and boundary identities give the scalar, mass,
and boundary inequalities in both deformations. The maximum estimate
controls finite slopes without differentiating the trace cutoff; the
energy and Newton-current estimates supply the seam-continuity argument
in Section~\ref{sec:transmission-analysis}.

Let $(U,g)$ be a smooth three-dimensional Riemannian manifold.  Unless a
metric is displayed, covariant derivatives, norms, contractions, volume
forms, and divergences in this Section use $g$.  For a symmetric covariant
tensor $C$, the notation $C p$ means the vector obtained by contracting with
$p$ and raising the remaining index with $g$.  For a positive smooth
function $N$ and a smooth function $f$, define $s>0$ and $p$ by
\begin{equation}
 p=s^2\nabla f,\qquad s^2+|\dd f|^2s^4=N^2,
 \qquad v=\frac pN,\qquad W=\frac Ns,
 \qquad \bar g=g+s^2\dd f\otimes\dd f.
 \label{eq:calculus-constitutive}
\end{equation}
The positive root is unique, and it depends smoothly on $(N,\dd f)$,
including at $\dd f=0$, because
\begin{equation}
 s^2=\frac{2N^2}{1+\sqrt{1+4N^2|\dd f|^2}}.
 \label{eq:calculus-positive-root}
\end{equation}
We use $d=s/N=\sqrt{1-|v|^2}$ for a scalar in $(0,1]$; differentials are
written $\dd$.  Direct rank-one inversion and the determinant formula give
\begin{equation}
 \bar g^{-1}=g^{-1}-v\otimes v,\qquad
 \dd V_{\bar g}=W\,\dd V_g.
 \label{eq:calculus-volume-inverse}
\end{equation}
For a further positive function $w$ put
\begin{equation}
 x=sw,\qquad h=x\bar g,\qquad A=Ns\bar g^{-1},
 \qquad \pi=N^{-1}\operatorname{sym}\nabla p^\flat,
 \label{eq:calculus-conformal-definitions}
\end{equation}
where $\operatorname{sym}$ is the half-sum.  The tensor $\pi$ throughout
this Section is the graph second fundamental form, not the electrostatic
stress introduced in Section~\ref{sec:stress-extension}.  The elementary
identities
\begin{equation}
 p\cdot\dd f=W^2-1,\qquad
 A(\dd f,\dd f)=W-W^{-1}\le p\cdot\dd f,
 \qquad 0<A\le N^2g^{-1}
 \label{eq:calculus-energy-algebra}
\end{equation}
will also be used in the penalty estimates.

\subsection{Curvature and the conserved current}

\begin{lemma}[Static graph curvature]\label{lem:graph-curvature}
Suppose that $\divg_g p=TN$, where $T$ is a smooth scalar function.
Then $\tr_g\pi=T$ and
\begin{align}
 \bar\nabla\log s&=\nabla\log N-\pi(v),
 \label{eq:calculus-lapse-differential}\\
 R_{\bar g}
 &=R_g+T^2-|\pi|^2-2(\divg_g\pi-\dd T)\cdot v.
 \label{eq:calculus-static-curvature}
\end{align}
More generally, let $\mathcal K$ be any smooth symmetric covariant tensor,
and define
\begin{equation}
 \kappa=\tr_g\mathcal K,\qquad
 \ell=T-\kappa,\qquad \lambda=\pi-\mathcal K,
 \qquad C=\lambda-\ell g,
 \label{eq:calculus-trace-definitions}
\end{equation}
and
\begin{equation}
 \mathscr D(v)=R_g+\kappa^2-|\mathcal K|^2
       -2(\divg_g\mathcal K-\dd\kappa)\cdot v.
 \label{eq:calculus-dominant-expression}
\end{equation}
Then
\begin{equation}
 R_{\bar g}=\mathscr D(v)+|\lambda|^2-\ell^2
                   -\frac2N\divg_g(Cp).
 \label{eq:calculus-completed-curvature}
\end{equation}
\end{lemma}

\begin{proof}
The trace assertion follows immediately from the definition of $\pi$.
To prove the first identity without any geometric interpretation, set
$a=\dd\log s$.  Since $p^\flat=s^2\dd f$,
\[
 \nabla_i p_j=N\pi_{ij}+a_i p_j-a_j p_i.
\]
Differentiating $s^2=N^2-|p|^2$ and substituting this formula gives
\[
 N^2a_i-(a\cdot p)p_i=N\nabla_iN-N\pi_{ij}p^j.
\]
Division by $N^2$ and Equation~\eqref{eq:calculus-volume-inverse} give
Equation~\eqref{eq:calculus-lapse-differential}, as an equality of vectors.

For the curvature identity consider, solely for this computation, the
Lorentzian metric
\[
 \widetilde g=-s^2\dd t^2+\bar g
\]
on $\R\times U$.  The graph $t=f$ has induced metric $g$.  If
$X^\mathrm{tan}=X+\dd f(X)\partial_t$, its future unit normal is
$n=(\partial_t+p^\mathrm{tan})/N$.  Thus
$\partial_t=Nn-p^\mathrm{tan}$.  The tangential part of the Killing
equation for $\partial_t$ is
$N\widetilde g(\widetilde\nabla_{X^\mathrm{tan}}n,Y^\mathrm{tan})
=\operatorname{sym}\nabla p^\flat(X,Y)$.
Its second fundamental form, with the sign convention specified after
\eqref{eq:future-trapping},
is therefore $\pi$.

Write $\widetilde G$ for the Einstein tensor and
$n_0=s^{-1}\partial_t$ for the normal to a static slice.  The contracted
Gauss and Codazzi identities with timelike unit normal and second form
$\widetilde g(\widetilde\nabla n,\cdot)$ read
\[
 2\widetilde G(n,n)=R_g+T^2-|\pi|^2,
 \qquad
 \widetilde G(n,X^\mathrm{tan})
       =(\divg_g\pi-\dd T)(X).
\]
The static slices have zero second form.  Applying these same contractions
to a static slice gives
$\widetilde G(n_0,n_0)=R_{\bar g}/2$ and
$\widetilde G(n_0,X)=0$ for spatial $X$.  The static time component of
$n$ is $N/s$, whereas that of $\partial_t/N$ is $s/N$.  Consequently
\[
 \frac12R_{\bar g}
 =\widetilde G(n,\partial_t/N)
 =\widetilde G(n,n)-\widetilde G(n,v^\mathrm{tan}),
\]
which proves Equation~\eqref{eq:calculus-static-curvature} with the
claimed sign of the momentum term.

For completeness, the completion of the square uses symmetry to give
\[
 \frac1N\divg_g(Cp)
 =(\divg_g\lambda-\dd\ell)\cdot v
                         +\lambda:\pi-\ell T.
\]
Also $2\ell T-\ell^2=T^2-\kappa^2$ and
$|\lambda|^2-2\lambda:\pi=|\mathcal K|^2-|\pi|^2$.
Substituting these two equalities in the right side of
Equation~\eqref{eq:calculus-completed-curvature} recovers
Equation~\eqref{eq:calculus-static-curvature} term by term.
\end{proof}

\begin{lemma}[Graph current and conformal curvature]\label{lem:graph-current}
Under the hypotheses and notation of Lemma~\ref{lem:graph-curvature}, set
\begin{equation}
 B=\nabla N-(\mathcal K+\ell g)p,\qquad
 \divg_g B=\Phi,\qquad -\divg_g(A\dd w)=\Psi,
 \qquad Q=B+A\dd w.
 \label{eq:calculus-general-system}
\end{equation}
Then
\begin{align}
 Q&=N\bar\nabla x+xCp-(x-1)B,\notag\\
 \divg_g Q&=\Phi-\Psi,
 \label{eq:calculus-general-current}\\
 N\Delta_{\bar g}x+x\divg_g(Cp)&=x\Phi-\Psi,
 \label{eq:calculus-general-x}\\
 xR_h&=\mathscr D(v)+|\lambda|^2-\ell^2
       -\frac{2\Phi}{N}+\frac{2\Psi}{Nx}
       +\frac32|\dd\log x|_{\bar g}^2.
 \label{eq:calculus-general-conformal}
\end{align}
In particular, the following two specializations hold.

\smallskip\noindent
\emph{The maximal AH system.}  If $\tr_gK=0$ and
\begin{equation}
 \divg_g p=0,\quad B=\nabla N-Kp,\quad
 \divg_g B=3N+\Sigma,\quad
 -\divg_g(A\dd w)=3N+\Sigma x+P,
 \label{eq:calculus-ah-system}
\end{equation}
then, writing
$D=R_g+6-|K|^2-2\divg_gK\cdot v$, one has
\begin{align}
 Q&=N\bar\nabla x+x(\pi-K)p-(x-1)B,\notag\\
 \divg_g Q&=-(x-1)\Sigma-P,
 \label{eq:calculus-ah-current}\\
 \left(\Delta_{\bar g}-3+
          N^{-1}\divg_g((\pi-K)p)\right)x&=-3-P/N,
 \label{eq:calculus-ah-x}\\
 x^2(R_h+6)
 &=x\bigl(D+|\pi-K|^2\bigr)
   +\frac{3}{2x}|\dd x|_{\bar g}^2
   +6(x-1)^2+\frac{2P}{N}.
 \label{eq:calculus-ah-curvature}
\end{align}
For the constraint densities in \eqref{eq:constraints},
$D=16\pi(\mu-J\cdot v)\ge0$ under the dominant energy condition.

\smallskip\noindent
\emph{The compact trace system.}  Let $0\le u_*\le1$ be constant and
take $\mathcal K=u_*K$, $\kappa=u_*\tr_gK$, $\ell=T-\kappa$.
If
\begin{equation}
 \begin{aligned}
 \divg_g p&=TN,&
 B&=\nabla N-(u_*K+\ell g)p,\\
 \divg_g B&=\Sigma_++\Sigma_-,&
 -\divg_g(A\dd w)&=\Sigma_+x+P,
 \end{aligned}
 \label{eq:calculus-compact-system}
\end{equation}
then
\begin{align}
 Q&=N\bar\nabla x+x(\lambda-\ell g)p-(x-1)B,\notag\\
 \divg_gQ&=-(x-1)\Sigma_+-P+\Sigma_-,
 \label{eq:calculus-compact-current}\\
 xR_h&=\mathscr D(v)+|\lambda|^2-\ell^2
             -\frac{2\Sigma_-}{N}+\frac{2P}{Nx}
             +\frac32|\dd\log x|_{\bar g}^2.
 \label{eq:calculus-compact-curvature}
\end{align}
Here $\lambda=\pi-u_*K$.  At $u_*=1$ this is the scalar identity for
the synthetic data.  It also holds on the outer region with
$p=T=K=\ell=0$, $\Sigma_-=0$, and $\mathscr D=R_{g_o}$.
\end{lemma}

\begin{proof}
Since $w=x/s$, Equation~\eqref{eq:calculus-lapse-differential} gives
\[
 A\dd w=N\bar\nabla x-Nx\bar\nabla\log s
        =N\bar\nabla x-x\nabla N+x\pi p.
\]
Adding $B$ proves the first formula for $Q$.  Its divergence from the
defining equations is $\Phi-\Psi$.  To compute the divergence of its
other expression, Equation~\eqref{eq:calculus-volume-inverse} gives
\begin{align*}
 \divg_g(N\bar\nabla x)
 &=N\Delta_{\bar g}x
       +N\bar g^{-1}(\dd\log s,\dd x)\\
 &=N\Delta_{\bar g}x+(\nabla N-\pi p)\cdot\dd x.
\end{align*}
The derivative-of-$x$ terms in $\divg_g(xCp+(1-x)B)$ are
$(Cp-B)\cdot\dd x=(\pi p-\nabla N)\cdot\dd x$.
They cancel exactly, leaving
\[
 \divg_gQ=N\Delta_{\bar g}x+x\divg_g(Cp)+(1-x)\Phi.
\]
This proves Equation~\eqref{eq:calculus-general-x}.  In dimension three
the conformal change $h=x\bar g$ has scalar curvature
\[
 xR_h=R_{\bar g}-2x^{-1}\Delta_{\bar g}x
                     +\frac32x^{-2}|\dd x|_{\bar g}^2.
\]
Substitution of Equations~\eqref{eq:calculus-completed-curvature} and
\eqref{eq:calculus-general-x} cancels the two occurrences of
$2N^{-1}\divg_g(Cp)$ and proves
Equation~\eqref{eq:calculus-general-conformal}.
The two stated systems now follow by substituting their values of
$T,\mathcal K,\Phi,\Psi$.  In the AH case, the nondifferential
cosmological terms after multiplication by $x$ are
$6x^2-12x+6=6(x-1)^2$.
Finally, $|v|<1$ and the constraint definitions give the asserted sign
of $D$.
\end{proof}

For later use, the divergence theorem makes these current formulas into
exact finite-domain flux budgets.  Let $S$ be the inner boundary, with
normal pointing into the domain, let $S_R$ have the outward normal, and
suppose $B_n=-b_S$ and $(A\dd w)_n=0$ on $S$.  The scalar $b_S$ is a
boundary datum, unrelated to the hyperbolic reference metric $b$.
In the compact construction assume also that the two currents have
matching normal flux at their common face, measured toward the end.
Then
\begin{equation}
 -\int_{S_R}Q_n\,\dd A=
 \begin{cases}
 \displaystyle\int_S b_S\,\dd A+
    \int_{U_R}\bigl[(x-1)\Sigma+P\bigr]\,\dd V,
       &\text{for \eqref{eq:calculus-ah-system}},\\[3pt]
 \displaystyle\int_S b_S\,\dd A+
    \int_{U_R}\bigl[(x-1)\Sigma_++P-\Sigma_-\bigr]\,\dd V,
       &\text{for \eqref{eq:calculus-compact-system}}.
 \end{cases}
 \label{eq:calculus-flux-budget}
\end{equation}
Indeed, the outward normal at $S$ is $-n$, so its contribution to the
outward flux is $\int_S b_S$.  The two face contributions cancel since
the base induced area forms and the directed fluxes agree.  Conversion
of the outer flux into the relevant mass difference requires the end
estimates proved in the corresponding construction.

\subsection{Boundary identities}

\begin{lemma}[Constant-height boundary]\label{lem:graph-boundary}
Let $\mathcal S$ be a smooth two-sided hypersurface on which $f$ is
constant.  Choose a $g$-unit normal $n$, and write $H=\divg_{\mathcal S}n$,
$\hat z=v\cdot n$, $y=\hat z^2$, and
$k_T=\tr_{\mathcal S}\mathcal K$.  The same directed normal is used on
each side if $\mathcal S$ is a gluing face.  Then
\begin{align}
 d^2\partial_n\log s&=B_n/N-k_T\hat z+yH,
 \label{eq:calculus-boundary-lapse}\\
 s\sqrt{x}\,H_h
 &=N(H-k_T\hat z)+B_n+(A\dd w)_n/x.
 \label{eq:calculus-boundary}
\end{align}
In particular, in the maximal AH system with $v=-z n$,
$k=K(n,n)$, $B_n=-b_S$, and $(A\dd w)_n=0$,
\begin{equation}
 s\sqrt{x}\,H_h=N(H-zk)-b_S=NH+\partial_nN.
 \label{eq:calculus-ah-boundary}
\end{equation}
In the compact system, if $H\le0$, $k_T<0$, $\hat z\le0$,
$B_n=-b_*<0$, and $(A\dd w)_n=0$, then $H_h<0$.

Suppose, at a compact gluing face, that $N_o=s_i$, that $w$ and the
normal fluxes of $B,A\dd w$ agree, that $f$ is constant on the inner
face, and that $p=0$ outside.  The induced $h$-metrics then agree.
If the inner base mean curvature satisfies $H_i>|k_T|$ and the outer
one is $H_o=\sqrt{H_i^2-k_T^2}$, then
$H_{h,i}\ge H_{h,o}$, both measured toward the end.
\end{lemma}

\begin{proof}
On $\mathcal S$, $p=N\hat z n$.  For tangent vectors $X,Y$,
$\nabla^2f(X,Y)=f_n\operatorname{II}(X,Y)$; consequently
$\pi(X,Y)=\hat z\operatorname{II}(X,Y)$ and
$\pi(n,n)=T-\hat z H$.  Taking the normal component in
Equation~\eqref{eq:calculus-lapse-differential} therefore yields
\[
 d^2\partial_n\log s
 =\partial_n\log N-\hat z T+yH.
\]
Since $\partial_nN=B_n+N(\mathcal K(n,n)+\ell)\hat z$ and
$\mathcal K(n,n)+\ell-T=-k_T$, this proves
Equation~\eqref{eq:calculus-boundary-lapse}.

In base normal coordinates, the tangential entries of $\bar g$ and
their first normal derivatives agree with those of $g$ at the face:
the extra products contain at least one tangential derivative of $f$.
The mixed entries vanish along the entire face, and the normal length
factor is $W$.  Hence $\bar n=dn$ and $H_{\bar g}=dH$.  The
mean-curvature formula for $h=x\bar g$ gives
\[
 \sqrt{x}\,H_h=d(H+\partial_n\log x).
\]
Multiplication by $s=Nd$, followed by
Equation~\eqref{eq:calculus-boundary-lapse}, gives
\[
 s\sqrt{x}\,H_h
 =N(H-k_T\hat z)+B_n+Nd^2w_n/w.
\]
The final term equals $(A\dd w)_n/x$, proving
Equation~\eqref{eq:calculus-boundary}.  In the maximal case
$k_T=-k$, $\hat z=-z$, and $\partial_nN=-Nzk-b_S$;
this proves Equation~\eqref{eq:calculus-ah-boundary}.
The asserted strict sign in the compact case follows because
$H-k_T\hat z\le0$ and $B_n<0$.

At the gluing face the tangential graph metrics equal their base
metrics, which agree, and $x_i=s_iw=N_ow=x_o$.
The remaining inequality reduces by
Equation~\eqref{eq:calculus-boundary} to
\[
 N_i(H_i-k_T\hat z)\ge s_i\sqrt{H_i^2-k_T^2}.
\]
Both sides are nonnegative, and after division by $N_i$ its squared
difference is
\[
 (H_i-k_T\hat z)^2-(1-\hat z^2)(H_i^2-k_T^2)
       =(k_T-H_i\hat z)^2\ge0.
\]
This proves the mean-curvature ordering.
\end{proof}

\subsection{A maximum estimate without differentiating the cutoff}

\begin{lemma}[Gradient maximum estimate]\label{lem:gradient-maximum}
Let $N>0,f,\mathcal K,T$ be smooth and satisfy
$\divg_g p=TN$ and
$\divg_g[\nabla N-(\mathcal K+\ell g)p]=\Phi$, with the definitions
above.  Suppose on the region in question that
\begin{equation}
 |\Ric_g|+|\mathcal K|+
       |\divg_g\mathcal K-\dd\tr_g\mathcal K|+|T|\le C_0,
 \qquad \Phi\le C_0N+\Sigma,\qquad \Sigma\ge0.
 \label{eq:calculus-maximum-hypotheses}
\end{equation}
At a point where $m=|\dd f|>0$, set
\begin{equation}
 \begin{aligned}
 l&=\log m,\quad u=\log N,\quad e=\nabla f/m,\quad y=|v|^2,\\
 q&=\frac{1-y}{1+y},\quad \beta=1+q,\quad
 \mathcal P=g^{-1}+(q-1)e\otimes e,\quad G=-\log s.
 \end{aligned}
 \label{eq:calculus-gradient-notation}
\end{equation}
Let $j$ be twice continuously differentiable with $j''>0$, with either
sign allowed for $j'$, and let $\eta$ be a twice continuously
differentiable localization function.  At an interior local maximum of
$G+j(f)+\eta$ with $y\ge1/2$,
\begin{equation}
 j''q m^2\le C\left[1+\frac\Sigma N
       +(1-y)^2(j'm)^2+|\dd\eta|^2+|\nabla^2\eta|\right],
 \label{eq:calculus-gradient-maximum}
\end{equation}
where $C$ depends only on $C_0$.  In particular, at that point,
\begin{equation}
 W^2\le\frac C{j''}\left[
     N^2+N\Sigma+(j')^2
       +N^2\bigl(|\dd\eta|^2+|\nabla^2\eta|\bigr)\right].
 \label{eq:calculus-gradient-weighted}
\end{equation}
No positive lower bound for $N$, no derivative bound for $T$, and no
second derivative bound for $N$ enter these two estimates.

At a constant-height boundary, with the notation of
Lemma~\ref{lem:graph-boundary}, the exact normal derivative is
\begin{equation}
 d^2\partial_n\bigl(G+j(f)+\eta\bigr)
   =-B_n/N+k_T\hat z-yH+j'\hat z/N+d^2\partial_n\eta.
 \label{eq:calculus-gradient-boundary}
\end{equation}
For the maximal AH inner boundary, if $H\le k$, $b_S\ge0$, and
$v=-zn$, this implies
\begin{equation}
 G_n=-\frac{yH+u_n}{1-y}
       =\frac{b_S/N+zk-z^2H}{1-z^2}
       \ge\frac{zk}{1+z}.
 \label{eq:calculus-ah-gradient-boundary}
\end{equation}
\end{lemma}

\begin{proof}
Use a $g$-orthonormal frame at the point with first vector $e$ and
transverse indices $a,b\in\{2,3\}$.  Put
$U_{ab}=\nabla^2_{ab}f/m$ and $C_T=T/z$, where $z=\sqrt y$.
Differentiation of the constitutive law gives
\begin{equation}
 \begin{aligned}
 Nm&=\frac z{1-y},\qquad
 q'=\frac{\dd q}{\dd(l+u)}=-\frac{4yq}{(1+y)^2},\\
 \mathcal P:\nabla^2f&=mC_T-\beta\nabla f\cdot\dd u,
 \qquad \tr U=C_T-ql_1-\beta u_1.
 \end{aligned}
 \label{eq:calculus-height-nondivergence}
\end{equation}
Here and below $q'$ differentiates the scalar constitutive function,
not a spatial coordinate.  The derivative of the height flux with
respect to $\dd f$ is $s^2\mathcal P$; its derivative with respect to
$N$ is $s^2(\beta/N)\nabla f$.  This also proves that the
nondivergence equation is smooth and elliptic through zero gradient.

Differentiating the height equation in direction $e$, commuting the
third derivatives of a scalar, and differentiating its norm gives
\begin{align}
 \mathcal P:\nabla^2l={}&\Ric(e,e)-\beta u_{11}
   -q'(l_1+u_1)^2-\beta\dd l\cdot\dd u
   +|U|^2+(1-q)|\dd_\perp l|^2-ql_1^2 \notag\\
 &+\frac{\partial_1T}{z}
        +C_T\bigl[(1-q)l_1-qu_1\bigr].
 \label{eq:calculus-log-gradient-equation}
\end{align}
Here is an explicit account of the norm and coefficient terms in this
calculation.  In $m^{-1}\mathcal P:\nabla^2m$, the second derivatives
of the norm contribute $q|\dd_\perp l|^2+|U|^2$.
The contraction $m^{-1}(\nabla_e\mathcal P):\nabla^2f$ is
$q'(l_1+u_1)l_1+2(q-1)|\dd_\perp l|^2$.
Differentiating $-\beta m u_1$ contributes
$-\beta u_{11}-\beta\dd l\cdot\dd u
-q'(l_1+u_1)u_1$.
Passing from $m$ to $l$ subtracts
$\mathcal P(\dd l,\dd l)=ql_1^2+|\dd_\perp l|^2$.
Finally,
$m^{-1}\partial_1(mT/z)=T_1/z+C_T[(1-q)l_1-qu_1]$.
These are precisely all terms of
Equation~\eqref{eq:calculus-log-gradient-equation}.

Set $I=(1+y)^{-1}$ and $J=y/(1+y)$.  Then
$G_l=J$, $G_u=-I$, and all four second partial derivatives of $G$
as a function of $(l,u)$ equal $-q'/2$.
Since $-I(q-1)=J\beta$, the Hessian terms in $u$ in
$\mathcal P:\nabla^2G$ reduce exactly to $-I\Delta u$.
The lapse equation, expanded using the symmetry of
$\mathcal K+\ell g$, is
\begin{equation}
 \Delta u=\Phi/N+
     [\divg_g\mathcal K-\dd\kappa+\dd T]\cdot v
     +(\mathcal K+\ell g):\pi-|\dd u|^2.
 \label{eq:calculus-log-lapse}
\end{equation}
The coefficient of $T_1$ is $J/z-Iz=0$.  Thus no derivative of $T$
remains.  More precisely, with $a_y=-q'/2$, the resulting identity is
\begin{align}
 \mathcal P:\nabla^2G={}&J\Ric(e,e)-I\Phi/N
   -I(\divg_g\mathcal K-\dd\kappa)\cdot v
   -I(\mathcal K+\ell g):\pi \notag\\
 &+\mathcal Q+JC_T[(1-q)l_1-qu_1],
 \label{eq:calculus-master-gradient}\\
 \mathcal Q={}&J|U|^2+a_y|\dd l+\dd u|^2
  -J\beta\dd l\cdot\dd u
  +J(1-q)|\dd_\perp l|^2-Jql_1^2+I|\dd u|^2.
 \label{eq:calculus-gradient-quadratic}
\end{align}
In obtaining the coefficient $a_y$ of the normal square, one uses
$2J+q=1$: the two contributions are $-Jq'$ and $-qq'/2$.

At the specified maximum put $D_0=j'm$.  The vanishing first derivative
is equivalent to
\begin{equation}
 J(l_i+u_i)=u_i-D_0\delta_{i1}-\eta_i.
 \label{eq:calculus-maximum-substitution}
\end{equation}
First take $\dd\eta=0$ and separate
$|U|^2=|U_{\rm tf}|^2+(\tr U)^2/2$.
The portion of
$\mathcal Q+j'\mathcal P:\nabla^2f$ independent of $C_T$ is exactly
\begin{align}
 J|U_{\rm tf}|^2
 &+\frac{(3-2y)u_1^2-6(1-y)D_0u_1
                         +3(1-y)^2D_0^2}{2y(1+y)} \notag\\
 &+\frac{2-y}{y(1+y)}|\dd_\perp u|^2.
 \label{eq:calculus-maximum-positive-form}
\end{align}
This follows by inserting
$l_1=(Iu_1-D_0)/J$, $l_a=Iu_a/J$, and
$\tr U=C_T+[-u_1+(1-y)D_0]/y$ in
Equation~\eqref{eq:calculus-gradient-quadratic}.
The terms containing $C_T$, including the last term of
Equation~\eqref{eq:calculus-master-gradient}, are
\begin{align}
 \frac J2 C_T^2+
 C_T[J(1-2q)l_1-J(\beta+q)u_1+D_0]
   =\frac J2C_T^2+qC_T(2D_0-u_1).
 \label{eq:calculus-trace-cancellation}
\end{align}
This last simplification is the gain needed for a bounded trace target:
$q|D_0|\le(1-y)|D_0|$, rather than an unweighted $|D_0|$.

For $y\in[1/2,1)$ the positive coefficients in
Equation~\eqref{eq:calculus-maximum-positive-form} are bounded below.
Young's inequality therefore bounds that expression below by
$c(|U_{\rm tf}|^2+|\dd u|^2)-C(1-y)^2D_0^2$.
The trace terms in Equation~\eqref{eq:calculus-trace-cancellation}
have the same lower bound after reducing $c$ and adding a constant,
because $|C_T|\le\sqrt2 C_0$.
For general $\dd\eta$, use $D_0+\eta_1$ in the normal substitutions.
The unchanged term $-\beta D_0u_1$ adds $\beta\eta_1u_1$ relative to
that substitution.  The transverse substitutions add terms bounded by
$C(|\dd\eta|^2+|\dd\eta||\dd u|)$, and the additional trace term is
$-(1-2q)C_T\eta_1$.  Another application of Young's inequality thus
adds only $C|\dd\eta|^2$ to the lower bound.

The components of the graph second form are
\begin{equation}
 \pi_{11}=T-z\tr U,\qquad \pi_{ab}=zU_{ab},\qquad
 \pi_{1a}=\frac{z\beta}{2}(u_a+l_a).
 \label{eq:calculus-second-form-components}
\end{equation}
Equation~\eqref{eq:calculus-maximum-substitution} also gives
\[
 |\tr U|\le C\bigl(1+|\dd u|+(1-y)|D_0|+|\dd\eta|\bigr).
\]
Thus the term $(\mathcal K+\ell g):\pi$ is absorbed in the same
positive quadratic expression, with an error
$C[1+(1-y)^2D_0^2+|\dd\eta|^2]$.
The curvature and momentum terms are bounded, and
$-I\Phi/N\ge-C-\Sigma/N$.
At a local maximum the positive definite tensor $\mathcal P$ has
nonpositive contraction with $\nabla^2(G+j(f)+\eta)$.
Its remaining height term is $j''\mathcal P(\dd f,\dd f)=j''qm^2$,
and $|\mathcal P:\nabla^2\eta|\le C|\nabla^2\eta|$.
This proves Equation~\eqref{eq:calculus-gradient-maximum}.

Multiply it by $N^2$ and use
$N^2qm^2=J/(1-y)$, $J\ge1/3$, and
$N^2(1-y)^2(j'm)^2=y(j')^2$.
This proves Equation~\eqref{eq:calculus-gradient-weighted} without a
lapse floor.  Lastly, negate
Equation~\eqref{eq:calculus-boundary-lapse} and use
$d^2f_n=\hat z/N$ to obtain
Equation~\eqref{eq:calculus-gradient-boundary}.
In the AH case, $\tr\pi=0$ also gives
$d^2\partial_n\log s=u_n+yH$.
Substitution of $u_n=-zk-b_S/N$ and $H\le k$ proves
Equation~\eqref{eq:calculus-ah-gradient-boundary}.
\end{proof}

\subsection{Energy and Newton flux identities at finite slope}

\begin{lemma}[Differential energy inequality]\label{lem:gradient-energy}
Assume the two differential equations and the geometric coefficient
bounds of Lemma~\ref{lem:gradient-maximum}.  On a region suppose, in
addition, that $N,s$ have positive lower bounds and finite upper bounds,
that $|\dd f|$ has a finite upper bound, and that $\Phi/N$ is bounded
above.  Constants in this Lemma may depend on these finite bounds.
With
$E_i=|\nabla N|^2+|\nabla^2f|^2$ and
$\rho=(1+y)/d$, one has
\begin{equation}
 \mathcal P:\nabla^2G\ge cE_i-C(1+|\dd G|^2).
 \label{eq:calculus-energy-nondivergence}
\end{equation}
There is a finite $k_0>0$, depending only on the stated bounds, such
that for $Y=s^{-k_0}$,
\begin{equation}
 \divg_g(\rho\mathcal P\dd Y)\ge cE_i-C.
 \label{eq:calculus-energy-divergence}
\end{equation}
These inequalities hold also at $\dd f=0$, and their constants use no
derivatives of $T$.

On a region where $p=0$, $s=N$, and $\Delta N=\Phi$, the corresponding
inequality is $\Delta Y\ge c|\nabla N|^2-C$.
At a constant-height gluing face satisfying $N_o=s_i$ and matching
flux of $B$, the directed inner flux $(\rho\mathcal P\dd Y)_n$ and
outer flux $\partial_nY$ differ by a bounded amount.  The bound
depends only on the finite field bounds and the fixed face geometry.
\end{lemma}

\begin{proof}
At nonzero gradient write $\gamma=\dd G$.  The identity
$\gamma=J\dd l-I\dd u$ gives
$\dd u=y\dd l-(1+y)\gamma$.
Inserting it into Equation~\eqref{eq:calculus-gradient-quadratic}
gives the exact expression
\begin{align}
 \mathcal Q={}&J\left[|U|^2+(1-y)l_1^2
                                  +(2-y)|\dd_\perp l|^2\right]\notag\\
 &-\frac{2y(2-y)}{1+y}\dd l\cdot\gamma
          +\frac{1+4y-y^2}{1+y}|\gamma|^2.
 \label{eq:calculus-energy-positive-form}
\end{align}
On the bounded field range, $1-y$ has a positive lower bound and
\[
 \frac J{m^2}=\frac{N^2(1-y)^2}{1+y},\qquad
 |\nabla^2f|^2=m^2(l_1^2+2|\dd_\perp l|^2+|U|^2).
\]
The first line of Equation~\eqref{eq:calculus-energy-positive-form}
therefore controls $|\nabla^2f|^2$.  Its mixed term is at most
$Cy|\dd l||\gamma|$ and is absorbed with an error $C|\gamma|^2$.
Moreover,
$|\dd u|^2\le C(y^2|\dd l|^2+|\gamma|^2)$, so after reducing the
positive coefficient the same lower bound controls $E_i$.

In Equation~\eqref{eq:calculus-master-gradient}, the trace term has
$|JC_T|\le C z$.  Its absolute value is bounded by
$C\sqrt{E_i}$, since $z$ is comparable to $m$ on this field range.
Equation~\eqref{eq:calculus-second-form-components}, or direct
differentiation of $p(N,\dd f)$, gives
$|\pi|\le C\sqrt{E_i}$.
The remaining terms in the master identity are bounded below by a
constant minus $C\sqrt{E_i}$.  Young's inequality proves
Equation~\eqref{eq:calculus-energy-nondivergence}.

There is no singularity at zero slope.  The expansion
$G=-\log N+\tfrac12N^2|\dd f|^2+O(|\dd f|^4)$ and the lapse equation
give, directly at a point with $\dd f=0$,
\[
 \mathcal P:\nabla^2G
 =N^2|\nabla^2f|^2+|\dd u|^2-\Phi/N
                    -N(\mathcal K+\ell g):\nabla^2f.
\]
This proves the same inequality there.  The matrices
$\mathcal P$ and $\rho\mathcal P$ are smooth functions of
$(N,\dd f)$, despite their expressions in terms of $e$; for example
$q-1=-2y/(1+y)$ cancels the denominator in $e\otimes e$.
Their derivatives consequently satisfy
$|\divg_g(\rho\mathcal P)|\le C\sqrt{E_i}$.

For $Y=e^{k_0G}$ the product rule yields
\begin{align*}
 \divg_g(\rho\mathcal P\dd Y)
 =k_0Y\bigl[\rho\mathcal P:\nabla^2G
      +\divg_g(\rho\mathcal P)\cdot\dd G
      +k_0\rho\mathcal P(\dd G,\dd G)\bigr].
\end{align*}
The mixed coefficient term is bounded by $C\sqrt{E_i}|\dd G|$.
Absorb half the positive $E_i$ term with Young's inequality, and then
choose $k_0$ so the final term absorbs the resulting multiple of
$|\dd G|^2$.  The upper and lower bounds on $s$ bound $Y$ on this
range and prove Equation~\eqref{eq:calculus-energy-divergence}.
Outside, direct differentiation gives
\[
 \Delta N^{-k_0}=k_0N^{-k_0}
          \left[(k_0+1)|\dd\log N|^2-\Phi/N\right],
\]
which proves the stated outer inequality.

On the face, $\rho q=d$ and
Equation~\eqref{eq:calculus-boundary-lapse} gives
\begin{equation}
 (\rho\mathcal P\dd Y)_n
   =-\frac{k_0YB_n}{s}
          +\frac{k_0Y}{d}(k_T\hat z-yH),
 \qquad
 (\partial_nY)_o=-\frac{k_0YB_n}{s}.
 \label{eq:calculus-energy-seam-flux}
\end{equation}
The second term in the inner expression is bounded under the stated
finite bounds; the first terms agree.  This proves the flux assertion.
\end{proof}

\begin{lemma}[Newton flux and freezing its coefficient]\label{lem:newton-flux}
In the setting of Lemma~\ref{lem:gradient-energy}, let
$\mathcal T_f=\nabla^2f-(\Delta f)g$.  The exact smooth vector identity
is
\begin{equation}
 \rho\mathcal P\nabla G
   =-\frac{\nabla N}{s}
      +\mathcal T_f\left(\frac{s^3}{N}\nabla f\right)
      +Ts\nabla f,
 \qquad
 \divg_g\mathcal T_f=\Ric_g(\nabla f,\cdot).
 \label{eq:calculus-newton}
\end{equation}
The coefficient of $\mathcal T_f$ is smooth at $\dd f=0$.

More specifically, in a fixed smooth face chart take compatible
reference fields
$\mathcal V_*=(n_*,a_*\dd t,s_*)$ for
$\mathcal V=(N_i,\dd f,N_o)$, where $t$ is inner signed unit normal
distance, $n_*>0$, and $s_*=s(n_*,|a_*|)$.
Assume the reference ranges and the actual fields lie in a fixed
bounded positive lapse and finite slope range.  Put
$Y_*=s_*^{-k_0}$ and
\[
 c_*=(k_0Y_*/s_*) ,\qquad
 \mathfrak b_*=k_0Y_*\frac{s_*^3}{n_*}a_*\nabla t.
\]
For the following assertion assume also that $\Phi$ is bounded
on each side.  Modify the inner and outer energy fluxes by
\begin{equation}
 \mathcal J_i=\rho\mathcal P\dd Y+c_*B-\mathcal T_f\mathfrak b_* ,
 \qquad \mathcal J_o=\nabla Y+c_*B.
 \label{eq:calculus-frozen-flux}
\end{equation}
They have bounded directed flux difference across the face, and
\begin{align}
 |\mathcal J|&\le C\bigl(1+|\mathcal V-\mathcal V_*|\sqrt E\bigr),
 \label{eq:calculus-frozen-flux-size}\\
 \divg\mathcal J&\ge c'E-C-C\delta_{\mathcal C}
 \label{eq:calculus-frozen-flux-divergence}
\end{align}
in the joined weak formulation, with $E=E_i$ inside and
$E=|\nabla N_o|^2$ outside.  The second line is understood with the
smooth metric densities in coordinates, and $\delta_{\mathcal C}$ is
surface measure on the face.  The same conclusion holds on an interior
chart with an arbitrary constant covector reference and no face term;
its metric-dependent reference coefficients are taken as smooth
functions of position.
\end{lemma}

\begin{proof}
At nonzero gradient the vector multiplying $\mathcal T_f$ can be
written as
\[
 \frac{y\nabla f}{d m^2}=
       \frac{s^3}{N}\nabla f,
 \qquad \frac{zT}{d}e=Ts\nabla f.
\]
For a transverse component, multiplication of the proposed identity
by $d$ gives $-u_a+yl_a$ on both sides.
For its first component, the two terms preceding $Ts\nabla f$ on the
right side, multiplied by $d$, are
\[
 -u_1-y\tr U=q(yl_1-u_1)-zT,
\]
by Equation~\eqref{eq:calculus-height-nondivergence}.
The last term adds $zT$, giving exactly the first component of the
left side.  This proves the first identity, including at zero gradient
by smoothness.  Commuting the covariant derivatives of $f$ gives
$\nabla^j\nabla^2_{ij}f=\nabla_i\Delta f+\Ric_{ik}\nabla^kf$,
which is the second identity.

Multiply the first identity by $k_0Y$.  In
Equation~\eqref{eq:calculus-frozen-flux}, the coefficients of
$\nabla N$ and of $\mathcal T_f$ then vanish at the reference state:
\begin{align*}
 \mathcal J_i={}&
   \left(c_*-\frac{k_0Y}{s}\right)\nabla N
   +\mathcal T_f\left(k_0Y\frac{s^3}{N}\nabla f-\mathfrak b_*\right)\\
 &+c_*(B-\nabla N)+k_0YTs\nabla f.
\end{align*}
The last line is bounded, since
$B-\nabla N=-(\mathcal K+\ell g)p$.
The two coefficient differences in the first line are Lipschitz in
the bounded field variables, so they are bounded by
$C|\mathcal V-\mathcal V_*|$.
On the outer side,
$\mathcal J_o=(c_*-k_0N_o^{-k_0-1})\nabla N_o$,
whose coefficient also vanishes at $N_o=s_*$.
This proves Equation~\eqref{eq:calculus-frozen-flux-size}.

The reference vector $\mathfrak b_*$ and its covariant derivative are uniformly
bounded in the fixed chart.  Equation~\eqref{eq:calculus-newton} gives
\[
 |\divg_g(\mathcal T_f\mathfrak b_*)|
 \le C(1+|\nabla^2f|).
\]
Adding $c_*B$ changes the divergence by the bounded scalar $c_*\Phi$.
Thus Equation~\eqref{eq:calculus-energy-divergence} and Young's
inequality give $\divg\mathcal J\ge c'E-C$ on each open side.
If reference coefficients vary smoothly with the metric in an interior
chart, their derivatives add only terms bounded by $C(1+\sqrt E)$;
these are absorbed in the same way.

Across the face, the energy flux difference is bounded by
Equation~\eqref{eq:calculus-energy-seam-flux}, and the added current has
matching normal flux.  The vector $\mathfrak b_*$ is normal to the face.  Its
Newton contraction there uses only
\[
 \mathcal T_f(n,n)=-\tr_{\mathcal C}\nabla^2f=-f_nH_i,
\]
which is bounded by the constant height trace and the slope bound.
Consequently the modified normal flux difference is bounded.  Integration
by parts on the two sides, whose face area densities agree, gives
the surface error in
Equation~\eqref{eq:calculus-frozen-flux-divergence}.
\end{proof}
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/03-graph-calculus.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/04-area-transfer.tex ====
\section{Finite-time area transfer}\label{sec:area-transfer}

The two deformations used below can contract a metric on a set of small
weighted volume.  The following argument shows that this contraction
cannot substantially decrease the minimum enclosing area.  Its constants
use a scalar-curvature lower bound and the topology of the original
exterior.  No curvature bound for the changing metrics, and no limit of a
hyperbolic Hawking mass, is needed.

\subsection{Caps, intrinsic boundaries, and the initial surface}

Attach a compact orientable handlebody to each component of the original
boundary $S$.  Write $O$ for the union of these caps and $X$ for the
resulting connected manifold without boundary.  This operation is
topological: whenever a comparison metric is given, extend it smoothly
across $S$ into the caps.  The end is unchanged, and the extension is
complete.  The topology of $X$ and $O$ will remain fixed.  In particular,
\begin{equation}\label{eq:flow-topology-constant}
 b_2=\dim_{\mathbb F_2}H_2(X\setminus\operatorname{int}O;\mathbb F_2)<\infty.
\end{equation}
Finiteness follows by retracting the product end onto its inner sphere;
what remains is a compact manifold with boundary.

We will start from a largest perimeter-minimizing enclosure $E_0$ of
$O$ alone.  In each application, the obstacle and end barriers establish
that its boundary is smooth, free, minimal, and disjoint from $O$.
Thus $O\subset E_0$ and $\partial O\subset\operatorname{int}E_0$.
The open representative $E_0$ is precompact.  Each component meets $O$,
since an extra component could be removed with a strict decrease of
perimeter.  Bounded components of its complement can similarly be
filled.  A largest minimizer is strictly outward minimizing: a larger
set with equal perimeter would also be a minimizer, contrary to
maximality.  This assertion concerns all precompact competitors, because
the far-out barriers allow any such competitor to be trimmed into the
fixed truncation used in constructing $E_0$.

The cap construction respects the intrinsic-boundary convention in
Section~\ref{sec:hypotheses}.  If $D$ is any admissible closed end-side
domain in the original exterior, then
$X\setminus\operatorname{int}_X D$ is a precompact finite-perimeter
enclosure of $O$.  Its perimeter is bounded by the area of the
\emph{entire} intrinsic boundary of $D$.  In particular, portions along
$S$ are counted; for $D=\Omega$ the perimeter is $|S|$, not zero.
Conversely, a smooth enclosure of $O$ whose boundary is disjoint from
$O$ and whose exterior is connected gives an admissible cut by taking
the closure of its exterior.  These observations also apply when the
surfaces or the caps have several components.

\begin{lemma}[Adjustment at the initial surface]\label{lem:flow-initial-adjustment}
Let $\widehat h$ be a smooth complete metric on $X$, and suppose $E_0$
has the properties just described.  If
$\Scal_{\widehat h}\ge-C_R$ on $X\setminus E_0$, then an arbitrarily
small, compactly supported smooth conformal change gives a metric
$\widetilde h$ for which
\begin{equation}\label{eq:flow-initial-adjustment}
 |\partial E_0|_{\widetilde h}=|\partial E_0|_{\widehat h},\qquad
 H_{\widetilde h}>0,\qquad
 \int_{\partial E_0}H_{\widetilde h}^{2}\,\dd A_{\widetilde h}\le1,
 \qquad \Scal_{\widetilde h}\ge-C_R-1
 \quad\hbox{on }X\setminus E_0.
\end{equation}
The set $E_0$ remains strictly outward minimizing.  The size of the
change may be chosen separately for every comparison metric.
\end{lemma}

\begin{proof}
In a two-sided signed-distance collar of $\partial E_0$, let $q$ be
positive on the exterior and choose
$\chi=q\zeta(q)$, where $\zeta$ is a nonnegative smooth cutoff equal to
one near zero.  Extend $\chi$ by zero beyond the collar.  Then
$\chi=0$ and $\partial_\nu\chi=1$ at the initial surface, while
$\chi\ge0$ outside $E_0$.  Set
$\widetilde h=e^{2a\chi}\widehat h$, with $a>0$.  The conformal
mean-curvature formula gives $H_{\widetilde h}=2a$ on $\partial E_0$.
Consequently its area is unchanged and its squared-mean-curvature
integral is $4a^2|\partial E_0|_{\widehat h}$.

Every outward competitor has its boundary outside $E_0$, where the
metric has only increased, so strict outward minimality is preserved.
Finally,
\begin{equation}\label{eq:flow-conformal-scalar}
 \Scal_{\widetilde h}
 =e^{-2a\chi}\bigl(\Scal_{\widehat h}
       -4a\Delta_{\widehat h}\chi
       -2a^2|\dd\chi|_{\widehat h}^2\bigr).
\end{equation}
Choose $a$ so that
$4a\|\Delta\chi\|_\infty+2a^2\|\dd\chi\|_\infty^2\le1$ and
$4a^2|\partial E_0|\le1$.  Further decreasing $a$ gives any prescribed
$C^2$ or tensor-comparison tolerance.  There is no need for a uniform
collar width or a uniform positive lower bound for the initial area.
This modification will be used only for the flow calculation; mass
inequalities are applied to the original comparison metric.
\end{proof}

\subsection{Weak flow and its topology}

We specify the weak-flow input.  Theorem~3.1 of
\citet{HI2001} applies on a smooth complete connected
boundaryless manifold possessing a proper locally Lipschitz weak
subsolution with precompact initial set.  It supplies a proper weak
solution from every nonempty smooth precompact open set.  In dimension
three, Regularity~1.3, Hull Property~1.4, and Growth Lemma~1.6 give
compact $C^{1,\gamma}$ level boundaries for $0<\gamma\le1/2$,
outward minimality, and
exponential area growth from a minimizing hull.  Smooth Start
Lemma~2.4 applies when that hull is strict and its smooth boundary has
$H>0$.  We use the $H^2$ inequality from Step~1 of the proof of
Growth Formula~5.7, Equation~(5.22): it holds for almost every pair of
positive endpoints and precompact levels, without a connectedness or
scalar-curvature-sign assumption.  This form suffices for a
distributional inequality in time.  Our smooth initial adjustment
and Smooth Start Lemma~2.4 supply its initial upper trace.  Thus no
restart from an arbitrary nonsmooth positive-time level is needed.

The end hypothesis in this existence theorem holds for all metrics
used here.  On an $\AH$ end the coordinate-sphere mean curvature is
$2+o(1)$ and $|\dd\log r|=1+o(1)$; on an $\AF$ end the corresponding
quantities are $2/r+o(r^{-1})$ and $r^{-1}+o(r^{-1})$.
Thus $u_\infty=c\log r$ with $0<c<2$ has
\begin{equation}\label{eq:flow-end-subsolution}
 \divg_{\widehat h}\left(
    \frac{\nabla u_\infty}{|\nabla u_\infty|}\right)
 \ge |\nabla u_\infty|
\end{equation}
beyond a sufficiently large sphere $r=R$.  Use
$v=\max\{c\log(r/R),0\}-1$, extended by $-1$ throughout the inner
region.  Its negative sublevel is a nonempty smooth precompact set,
bounded by $r=R e^{1/c}$, and outside this set the displayed inequality
holds smoothly.  Extending the radial unit normal field by zero
inside $r=R$ gives a nonnegative distributional divergence
contribution at that sphere.  The Gauss--Green inequality against
outward variations proves the weak subsolution property as well.
This subsolution is proper toward infinity.  The compact
metric adjustment in Lemma~\ref{lem:flow-initial-adjustment} leaves it
unchanged.

For positive-time notation, extend the exterior level function $u$ by
zero on the initial set and, for $t>0$, write
\begin{equation}\label{eq:flow-levels}
 E_t=E_0\cup\{u<t\},\qquad N_t=\partial E_t,\qquad
 A(t)=|N_t|_{\widetilde h}=e^t A(0).
\end{equation}
Use the open perimeter representatives of these sets.  At almost every
time their boundary is $C^{1,\gamma}$, its weak mean curvature is
$H=|\nabla u|$, and its area measure agrees with the level measure in
the coarea formula.  Indeed, differentiability and nonvanishing
gradient hold at almost every point of almost every level in the
coarea sense; those points belong to the reduced boundary of the
sublevel set.  The equivalent statement follows directly from the BV
coarea formula.  The variational property used below is local
minimality of
\begin{equation}\label{eq:flow-variational-functional}
 F\longmapsto \Per_{\widetilde h}(F)
       -\int_F|\nabla u|_{\widetilde h}\,\dd V_{\widetilde h}
\end{equation}
under compact changes away from the initial set; differences of this
expression are taken in a compact region containing the change.

\begin{lemma}[A component bound on fixed topology]\label{lem:hull-components}
At almost every positive time, every component of $E_t$ meets $E_0$,
the open exterior of $E_t$ is connected, and
\begin{equation}\label{eq:flow-component-bound}
 b_0(N_t)\le b_2,\qquad \chi(N_t)\le2b_2,
\end{equation}
where $b_2$ is the fixed number in
\eqref{eq:flow-topology-constant}.
\end{lemma}

\begin{proof}
Fix a time $t$ with the stated regularity and with
$\Vol\{u=t\}=0$.  Excluding the latter exceptional times removes at
most a countable set.  Suppose a component $B$ of $E_t$ did not meet
$E_0$.  For almost every $0<s<t$, the set $E_s\cap B$ is compactly
contained in $B$ and separated from $E_0$.  To check this, continuity
gives $u\le s<t$ on its closure, and a neighborhood of
$\overline E_0$ belongs to $E_t$.  A connected component of that
neighborhood cannot meet $B$ without placing part of $E_0$ in $B$.
The boundaries here are perimeter representatives; the same
containment holds modulo null sets.

Removing $E_s\cap B$ is consequently an allowed variation in
\eqref{eq:flow-variational-functional}.  If
$p(s)=\Per_{\widetilde h}(E_s;B)$, minimality and coarea give
\begin{equation}\label{eq:flow-no-new-components}
 p(s)\le\int_{E_s\cap B}|\nabla u|\,\dd V
       =\int_0^s p(\ell)\,\dd\ell.
\end{equation}
There is no initial contribution because $B\cap E_0=\varnothing$.
Gr\"onwall's inequality implies $p=0$ almost everywhere.  A
finite-perimeter indicator with zero variation on the connected open
set $B$ is constant there.  Compact containment rules out the value
one, so $E_s\cap B$ is null for almost every $s<t$.  Letting such $s$
increase to $t$ contradicts that $B$ is a nonempty open component of
$E_t$.  This proves the first assertion.  It also shows that each such
component contains an entire component of $E_0$, and hence meets $O$.

Outward minimality excludes a bounded open component of the
complement: filling it removes its nonempty boundary and strictly
decreases perimeter.  Because $X$ has just one product end and $E_t$
is precompact, only one component of the exterior can be unbounded.
Its open exterior is therefore connected.

Let $N_t^1,\ldots,N_t^m$ be the boundary components.  Choose a large
compact truncation
$Y\subset X\setminus\operatorname{int}O$ containing all of them.
For each $j$, an arc can start on $\partial O$, travel in the adjacent
component of $E_t$, cross $N_t^j$ once transversely, and then travel in
the connected exterior to the outer boundary of $Y$, without meeting
any other $N_t^k$.  To construct its inner part, join a point just
inside $N_t^j$ to $O$ in the relevant connected open component and
stop at the first cap encountered; thus the arc stays outside the cap
interiors.  Small perturbations give the asserted transversality.

These arcs define relative one-cycles in $(Y,\partial Y)$ whose
mod-two intersection pairings with the closed surfaces $N_t^k$ are
the separate coordinate vectors.  Hence the classes $[N_t^k]$ are
linearly independent in $H_2(Y;\mathbb F_2)$.  Extending the
truncation along its product end changes no homology, so $m\le b_2$.
Each closed orientable surface has Euler characteristic at most two,
which proves the second estimate.  The arcs and the intersection
pairings also make explicit why disconnected caps and arbitrary
compact interior topology cause no difficulty.
\end{proof}

\begin{lemma}[Curvature control up to a prescribed area]\label{lem:flow-curvature}
Suppose the adjusted metric satisfies
$\Scal_{\widetilde h}\ge-C_0$ outside $E_0$, and let $A_*>0$.
At almost every time for which $A(t)\le A_*$,
\begin{equation}\label{eq:flow-uniform-curvature}
 J(t):=\int_{N_t}|\nabla u|^2\,\dd A_{\widetilde h}
 \le C_H:=1+\frac23 C_0 A_*+16\pi b_2.
\end{equation}
In particular this bound is independent of $A(0)$, even if the
initial areas tend to zero in a family.
\end{lemma}

\begin{proof}
Let $\mathcal A_t$ denote the weak second fundamental form of $N_t$.
For almost every pair $0<r<t$, the cited growth formula, after
dropping its nonnegative tangential gradient term, says
\begin{equation}\label{eq:flow-hi-growth}
 J(r)\ge J(t)+\int_r^t\int_{N_s}
 \bigl(2|\mathcal A_s|^2+2\Ric(\nu,\nu)-H^2\bigr)
 \,\dd A\,\dd s.
\end{equation}
After our adjustment, Smooth Start Lemma~2.4 identifies the weak
flow with the smooth flow on an initial interval.  Hence
$J(r)\to J(0)\le1$ as $r\downarrow0$.  Letting admissible
Lebesgue endpoints tend to zero supplies the initial upper trace in
the integrated inequality below.  At positive times only the
almost-everywhere inequality is used.

For completeness, Gauss and Gauss--Bonnet apply to these levels with
their weak curvature.  A local $C^{1,\gamma}$ graph with bounded weak
mean curvature belongs to $W^{2,2}$ by difference quotients in its
uniformly elliptic mean-curvature equation.  Smoothing graph
representations gives smooth embeddings converging in
$C^1\cap W^{2,2}$; the graph formula for the second fundamental form
then gives convergence of the quadratic curvature terms in $L^1$.
Thus the smooth Gauss identity and Gauss--Bonnet pass to the limit.
This is also the content of Lemmas~5.4--5.5 of
\citet{HI2001}.

The Gauss equation and $|\mathcal A_s|^2\ge H^2/2$ give
\begin{align}\label{eq:flow-gauss-bound}
 \int_{N_s}\bigl(2|\mathcal A_s|^2+2\Ric(\nu,\nu)-H^2\bigr)
 &=\int_{N_s}\bigl(|\mathcal A_s|^2+
                   \Scal_{\widetilde h}-\Scal_{N_s}\bigr)\notag\\
 &\ge\tfrac12J(s)-C_0A(s)-8\pi b_2.
\end{align}
Consequently, in distributions in time,
$J'+J/2\le C_0A+8\pi b_2$, with initial upper value at most one.
One can see this directly from \eqref{eq:flow-hi-growth} at Lebesgue
endpoints: $J$ plus the integral of the lower bound in
\eqref{eq:flow-gauss-bound} has a nonincreasing representative.  Any
singular part of its derivative is nonpositive, so the integrating
factor is valid.  It yields
\begin{equation}\label{eq:flow-integrated-curvature}
 J(t)\le e^{-t/2}
       +\int_0^t e^{(s-t)/2}\bigl(C_0A(s)+8\pi b_2\bigr)\,\dd s.
\end{equation}
Using $A(s)=e^{s-t}A(t)$ bounds the two integrals by
$\frac23C_0A(t)$ and $16\pi b_2$, respectively.
\end{proof}

\subsection{Projecting enclosing surfaces}

In the next lemma, $A_*$ is the original minimum enclosing area,
defined using complete intrinsic boundaries.  The comparison end may
be the original $\AH$ end, or the $\AF$ extension of a fixed compact
inner region.  In the latter case $g_{\mathrm{base}}$ denotes $g$ on
the inner region and $g_o$ on the extension.  Statements involving
this base metric are interpreted separately on the two sides of the
joining sphere $\mathcal C$.

\begin{lemma}[Enclosing-area projection]\label{lem:area-projection}
Let $N$ be a compact embedded $C^1$ enclosure of $O$, disjoint from
$S$, with connected exterior.
\begin{enumerate}[label=\textup{(\roman*)}]
\item On the original $\AH$ exterior, fix a large radius $r_p$ and put
\begin{equation}\label{eq:flow-ah-weight}
 \rho=\min\{1,r_p^2/r^2\},
\end{equation}
with $\rho=1$ on the compact part.  There is $q(r_p)\downarrow0$,
depending only on the original end metric, such that
\begin{equation}\label{eq:flow-ah-projection}
 \int_N\rho\,\dd A_g\ge(1-q(r_p))A_*.
\end{equation}
\item Suppose the $\AF$ extension has the parallel-surface form
$g_o=U^2\dd t^2+\gamma_t$, where $\gamma_t$ is induced on outward
Euclidean parallels of a strictly convex embedding of
$(\mathcal C,\gamma_0)$.  Then, with $\rho=1$,
\begin{equation}\label{eq:flow-af-projection}
 \int_N\dd A_{g_{\mathrm{base}}}\ge A_*.
\end{equation}
For a surface crossing the joining sphere, the assertion holds for
the piecewise area; it is enough that its intersection with the
joining sphere have zero two-dimensional measure.  In particular
this holds for almost every weak-flow level.
\end{enumerate}
\end{lemma}

\begin{proof}
First take $N$ smooth and transverse to the spheres used in the
construction.  Let $D$ be its closed end-side domain.  For part
\textup{(i)}, choose a transverse sphere $r=r_1$ with
$r_p/2\le r_1\le r_p$, and replace $D$ by
\begin{equation}\label{eq:flow-projected-domain}
 D'=D\cup\{r\ge r_1\}.
\end{equation}
This domain is connected because both sets contain the entire remote
end.  Its boundary inside $r_1$ is part of $N$; the remaining boundary
is a portion of the sphere $r=r_1$.  Every point of this newly exposed
sphere portion is covered by radial projection of $N\cap\{r\ge r_1\}$:
the ray starts outside $D$ and eventually lies in its end, so must
cross $N$.

In the hyperbolic reference metric the radial map from radius $r$ to
$r_1$ has two-dimensional Jacobian at most $r_1^2/r^2$ on every
two-plane.  The original metric has uniform relative error $o(1)$
on $r\ge r_p/2$.  Its corresponding Jacobian is therefore at most
\begin{equation}\label{eq:flow-projection-jacobian}
 (1+o_{r_p}(1))\frac{r_1^2}{r^2}
 \le(1+o_{r_p}(1))\rho.
\end{equation}
The area formula, allowing multiplicity, bounds the new sphere area
by the weighted area of the discarded portion of $N$.

The boundary of $D'$ has transverse corners along finitely many
smooth curves.  In local coordinates given by defining functions for
the two faces, round each quadrant edge in a shrinking tubular
neighborhood, toward the side enlarging $D'$.  The added area tends
to zero: the modification has vanishing width along a fixed compact
edge, with bounded local slopes.  The rounded end-side domain stays
connected and gives an admissible cut, disjoint from $S$.  Since the
definition of $A_*$ uses its full intrinsic boundary, it follows that
$A_*\le(1+o_{r_p}(1))\int_N\rho\,\dd A_g$.  This proves
\eqref{eq:flow-ah-projection}.

For part \textup{(ii)}, keep the part of $D$ in the original compact
inner region and attach the whole original $\AH$ region outside
$\mathcal C$.  The resulting end-side domain is connected.  Indeed,
every point of its inner part can be joined inside the old $D$ to
infinity, and the initial portion of such a path reaches
$\mathcal C$.  Its newly exposed boundary in $\mathcal C$ is covered
by projection of the old outer boundary along the parallel rays.
This projection has two-dimensional Jacobian at most one:
$\gamma_t\ge\gamma_0$ by strict convexity, its normal derivative is
zero, and the positive coefficient $U^2$ creates no mixed terms.
Rounding the resulting transverse corners as above gives admissible
original cuts with area at most
$\int_N\dd A_{g_{\mathrm{base}}}$ plus a vanishing error.

Finally, a $C^1$ embedded enclosing surface can be approximated by
smooth embeddings in $C^1$, within a collar disjoint from $S$, by an
ambient isotopy.  This preserves the end side and its connectedness.
Choose the approximations transverse to the cutting sphere.  Areas
and weighted areas converge.  In the joined case, first approximate
on the two smooth sides and make the approximations transverse to
$\mathcal C$; round across a shrinking seam neighborhood.  The base
metric is bounded and smooth on either side at this fixed
construction, and the surface has zero area in the seam.  The area
of its portion in that neighborhood tends to zero, so these local
roundings contribute a vanishing area error.  This procedure also
applies when the smooth collar coordinates chosen for the comparison
metric differ from the original side coordinates: the
identifications are smooth on each closed side and preserve the
joining surface.  For almost every flow level the zero-area
intersection follows by applying the level-measure decomposition to
the fixed finite-area seam.  Passing to the limit proves both
assertions.
\end{proof}

\subsection{The uniform transfer criterion}

The following formulation isolates all requirements on the
deformations.  A reference metric may be piecewise smooth across one
fixed compact joining surface, as in
Lemma~\ref{lem:area-projection}\textup{(ii)}.  Its area and volume
measures below are then the sums of the two side measures.

\begin{proposition}[Uniform area transfer]\label{prop:area-transfer}
Fix the capped topology $(X,O)$, a number $A_*>0$, and a scalar lower
bound $C_R\ge0$.  Suppose a comparison has the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The smooth complete metric $\widehat h$ has one $\AH$ or $\AF$
end with the mean-curvature and radial-gradient asymptotics used in
\eqref{eq:flow-end-subsolution}.  The cap-only enclosure $E_0$ is a
smooth precompact strictly outward-minimizing set, with minimal
boundary disjoint from $O$, every component meeting $O$, and
$\Scal_{\widehat h}\ge-C_R$ outside $E_0$.
\item On $X\setminus O$ there are a reference metric
$g_{\mathrm{base}}$, positive functions $x,W$, a metric $\bar g$, and
a weight $0<\rho\le1$, such that
\begin{equation}\label{eq:flow-metric-comparison}
 \bar g\ge g_{\mathrm{base}},\qquad
 \dd V_{\bar g}=W\,\dd V_{g_{\mathrm{base}}},\qquad
 (1-\eta)x\bar g\le\widehat h\le(1+\eta)x\bar g.
\end{equation}
\item Every enclosing $C^1$ surface with connected exterior, as in
Lemma~\ref{lem:area-projection}, satisfies
\begin{equation}\label{eq:flow-abstract-projection}
 \int_N\rho\,\dd A_{g_{\mathrm{base}}}\ge(1-q)A_*.
\end{equation}
In the piecewise case this is required for surfaces having zero area
in the joining surface.
\end{enumerate}
For every $0<\eps<1$ there are positive tolerances
$\eta_0,q_0,\delta_0$ and $v_0$, depending only on
$\eps,A_*,C_R$ and the integer $b_2$ in
\eqref{eq:flow-topology-constant}, with the following consequence.
If $\eta\le\eta_0$, $q\le q_0$, $0<\delta\le\delta_0$, and
\begin{equation}\label{eq:flow-bad-volume}
 \int_{\{x<1-\delta\}}\rho^{3/2}W\,\dd V_{g_{\mathrm{base}}}<v_0,
\end{equation}
then
\begin{equation}\label{eq:flow-area-conclusion}
 |\partial E_0|_{\widehat h}\ge(1-\eps)A_*.
\end{equation}
The tolerances are uniform over all comparisons satisfying these
conditions; the geometry and initial area of $E_0$ may vary.
\end{proposition}

\begin{proof}
Make the change from Lemma~\ref{lem:flow-initial-adjustment}.  It
preserves the initial area and all topological properties.  Take its
size sufficiently small that the comparison
\eqref{eq:flow-metric-comparison} for the resulting $\widetilde h$
holds with an error $\bar\eta\le\eps/32$, by initially requiring,
for example, $\eta\le\eta_0=\eps/64$.  Its scalar lower bound is
$-C_0$, where $C_0=C_R+1$.  Also take
$q_0=\delta_0=\eps/32$.

Suppose, to the contrary, that $A(0)<(1-\eps)A_*$.  Exponential area
growth gives an interval $I=[t_-,t_+]$ on which the endpoint areas
are $(1-\eps)A_*$ and $(1-\eps/2)A_*$.  Its length is the fixed
positive number
\begin{equation}\label{eq:flow-time-window}
 \Delta_\eps=\log\frac{1-\eps/2}{1-\eps}.
\end{equation}
Although $t_-$ may tend to infinity when $A(0)$ tends to zero,
Lemma~\ref{lem:flow-curvature} bounds $J(t)$ throughout $I$ by the
same constant $C_H$.

Write $\mathcal B=\{x<1-\delta\}$.  On the good part of a level,
\eqref{eq:flow-metric-comparison} and $\bar g\ge g_{\mathrm{base}}$
give
\begin{equation}\label{eq:flow-good-area}
 \int_{N_t\setminus\mathcal B}\rho\,\dd A_{g_{\mathrm{base}}}
 \le\frac{A(t)}{(1-\bar\eta)(1-\delta)}.
\end{equation}
Combine this with \eqref{eq:flow-abstract-projection}.  Since
$(1-\bar\eta)(1-\delta)\ge1-\eps/16$, for almost every $t\in I$
we obtain
\begin{equation}\label{eq:flow-bad-area-lower}
 \int_{N_t\cap\mathcal B}\rho\,\dd A_{g_{\mathrm{base}}}
 \ge (1-q)A_*
       -\frac{(1-\eps/2)A_*}{(1-\bar\eta)(1-\delta)}
 \ge \frac{\eps A_*}{4}=:c_A.
\end{equation}
The same metric comparison on the bad part therefore implies
\begin{equation}\label{eq:flow-weighted-area-lower}
 \int_{N_t\cap\mathcal B}\rho x^{-1}\,\dd A_{\widetilde h}
 \ge(1-\bar\eta)c_A\ge c_A/2.
\end{equation}

Here is the precise weighted coarea step.  Put $F=\rho x^{-1}$ and
$H=|\nabla u|_{\widetilde h}$.  For almost every level $H>0$ at
almost every point in its area measure.  H\"older with exponents
$3/2$ and $3$ gives
\begin{align}\label{eq:flow-holder-exponents}
 \int_{N_t\cap\mathcal B}F\,\dd A
 &=\int_{N_t\cap\mathcal B}
       \left(\frac{F^{3/2}}H\right)^{2/3}(H^2)^{1/3}\,\dd A\notag\\
 &\le
 \left(\int_{N_t\cap\mathcal B}\frac{F^{3/2}}H\,\dd A\right)^{2/3}
 J(t)^{1/3}.
\end{align}
Infinite integrals cause no difficulty; otherwise this identity can
first be applied where $H$ is bounded away from zero and passed to
the limit.  Equations~\eqref{eq:flow-uniform-curvature} and
\eqref{eq:flow-weighted-area-lower} yield
\begin{equation}\label{eq:flow-level-volume-lower}
 \int_{N_t\cap\mathcal B}
       \frac{\rho^{3/2}x^{-3/2}}{|\nabla u|}\,\dd A_{\widetilde h}
 \ge \frac{(c_A/2)^{3/2}}{\sqrt{C_H}}=:c_V.
\end{equation}
Integrating over $I$ and applying Lipschitz coarea gives
\begin{align}\label{eq:flow-coarea-contradiction}
 \Delta_\eps c_V
 &\le\int_{\mathcal B\cap\{t_-<u<t_+\}\cap\{|\nabla u|>0\}}
                  \rho^{3/2}x^{-3/2}\,\dd V_{\widetilde h}\notag\\
 &\le(1+\bar\eta)^{3/2}
       \int_{\mathcal B}\rho^{3/2}W\,\dd V_{g_{\mathrm{base}}}.
\end{align}
The last inequality uses
$\dd V_{\widetilde h}\le
(1+\bar\eta)^{3/2}x^{3/2}W\,\dd V_{g_{\mathrm{base}}}$.
Thus it is enough to take
$v_0=\Delta_\eps c_V/2^{3/2}$.
Notice that no contribution from a plateau of the level function
has been assumed: coarea integrates only where its gradient is
nonzero, and the full bad-set volume is an upper bound.  This proves
the proposition.
\end{proof}

\subsection{The two applications and their parameter order}

For the maximal $\AH$ reduction, the base is the original $g$ and
\begin{equation}\label{eq:flow-graph-volume}
 \bar g=g+s^2\dd f^2,\qquad h=x\bar g,\qquad
 W^2=1+s^2|\dd f|_g^2,\qquad
 \dd V_{\bar g}=W\,\dd V_g.
\end{equation}
The curvature bound is $\Scal_h\ge-6$, and the cap-only largest hull
is the enclosure used in that reduction.  Fix the area tolerance
$\eps$, then $r_p$ so that the error in
\eqref{eq:flow-ah-projection} is at most $q_0$, and then
$\delta\le\delta_0$.  The quantitative estimates of
Section~\ref{sec:maximal-reduction} give, with
$R_s=M^{3/10}$ and $L=M^2$,
\begin{equation}\label{eq:flow-ah-bad-volume}
 \int_{\{x<1-\delta\}}\rho^{3/2}W\,\dd V_g
 \le C\left(L^{-1}+\sqrt{M/L}
              +R_s^{-1}+\frac{\sqrt M}{R_s^2}\right)
 \longrightarrow0.
\end{equation}
To explain the powers, the unweighted bad volume inside $R_s$ is
$O(L^{-1})$, and $\int(W^2-1)\,\dd V_g\le CM$; Cauchy--Schwarz
gives the first two terms.  On the tail the $\AH$ volume element is
comparable to $r\,\dd r\,\dd A_\sigma$ and
$\rho^{3/2}=r_p^3/r^3$.  Hence
$\int_{r>R_s}\rho^{3/2}\,\dd V_g=O(R_s^{-1})$, whereas
$\int_{r>R_s}\rho^3\,\dd V_g=O(R_s^{-4})$; another
Cauchy--Schwarz inequality gives the last term.  The constants here
may depend on the already fixed $r_p$ and original data.
Proposition~\ref{prop:area-transfer} therefore yields the required
lower bound for the comparison boundary area.

For the compact construction with the $\AF$ attachment,
$g_{\mathrm{base}}=g$ inside and $g_o$ outside,
$\bar g=g+s^2\dd f^2$ inside and $\bar g=g_o$ outside.  Thus $W=1$
outside, $\rho=1$, and Lemma~\ref{lem:area-projection} has zero
projection error.  Section~\ref{sec:scalar-error} constructs a smooth
metric $h'$ arbitrarily close to $h=x\bar g$, with a uniform scalar
lower bound, together with two enclosures.  The enclosure $E$
contains the caps and all unsafe balls and is used for the
$\AF$ mass inequality.  The largest cap-only minimizer $E_0$ is
used for the flow.  In particular
\begin{equation}\label{eq:flow-af-two-enclosures}
 |\partial E|_{h'}\ge|\partial E_0|_{h'}.
\end{equation}
No topological bound on the unsafe balls or on $E$ is required.

The bad set is confined to a fixed outer radius by the capacitary
lower bound for $x$.  On its designated penalized region,
\begin{equation}\label{eq:flow-af-bad-volume}
 \int_{\{x<1-\delta\}\cap\mathrm{designated}}
                       W\,\dd V_{g_{\mathrm{base}}}\le C/L.
\end{equation}
The omitted fixed exterior collar has $W=1$, so its contribution is
at most its base volume.  The omitted inner collar has base width
$O(M^{-4})$.  Since the compact construction gives
$\int_{\Omega_i}W^2\,\dd V_g\le C(L)M^2$, its contribution is at
most
\begin{equation}\label{eq:flow-inner-collar-cost}
 O(M^{-2})\left(\int_{\Omega_i}W^2\,\dd V_g\right)^{1/2}
 =O_L(M^{-1}).
\end{equation}
Choose the exterior collar sufficiently thin and then $L$
sufficiently large, both before $M$.  Next choose $M$ large and the
remaining finite penalty parameters as in
Section~\ref{sec:compact-construction}.  Finally choose the individual
corner-smoothing tolerance small enough for
\eqref{eq:flow-metric-comparison}.  These choices make the bad volume
smaller than $v_0$.  The flow-only initial adjustment does not change
$|\partial E_0|_{h'}$.  Equations~\eqref{eq:flow-area-conclusion} and
\eqref{eq:flow-af-two-enclosures} give
\begin{equation}\label{eq:flow-af-area-conclusion}
 |\partial E|_{h'}\ge|\partial E_0|_{h'}\ge(1-\eps)A_*.
\end{equation}
If $A_*=0$ there is no area-transfer assertion to prove; the
nonnegative mass conclusion from the $\AF$ comparison supplies the
needed zero-area bound.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/04-area-transfer.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/05-maximal-reduction.tex ====
\section{Reduction of maximal data to a minimal boundary}
\label{sec:maximal-reduction}

The result of this section is a reduction within the asymptotically
hyperbolic class.  In particular, it requires no uniform statement about
the metrics to which the time-symmetric inequality is applied.

\begin{theorem}[Maximal reduction]\label{thm:ah-reduction}
Let \(F:[0,\infty)\to[0,\infty)\) be continuous and nondecreasing.
Suppose that every smooth, complete, connected, orientable, one-ended
asymptotically hyperbolic three-dimensional exterior with nonempty
compact minimal boundary, scalar curvature at least \(-6\),
\(g-b\in C^{2,\gamma}_\tau\) for some \(0<\gamma<1\) and
\(3/2<\tau<3\), the weighted scalar integrability of
Section~\ref{sec:hypotheses}, and four finite mass fluxes satisfies
\[
 p_0\ge F(A_{\min}).
\]
The exponent \(\gamma\) may depend on the comparison metric.
This hypothesis is to hold in every designated asymptotic chart,
without a causal-character assumption on the mass covector.
Then every maximal initial-data exterior of
Section~\ref{sec:hypotheses} satisfies, in each designated chart,
\[
 p_0(g)\ge F\bigl(A_{\min}(S;g)\bigr).
\]
\end{theorem}

Fix the original data and chart, and put \(A_*=A_{\min}(S;g)\).
Extend the end radius \(r\) to a smooth positive function bounded away
from zero on \(\Omega\).  Constants may depend on this extension and
the fixed data.  Derivatives, volume measures, and fluxes in this
section use \(g\), unless a different metric is indicated.
The letter \(b_S\) below denotes a scalar inner flux datum; it is
unrelated to the hyperbolic reference metric \(b\).

\subsection{The equations and the order of choices}

For \(N>0\) and \(f\), define
\begin{equation}\label{eq:ah-fields}
 \begin{gathered}
 s^2+s^4|df|_g^2=N^2,\qquad p=s^2\nabla f,\qquad
 v=p/N,\qquad W=N/s,\\
 \bar g=g+s^2df^2,\qquad
 \pi=N^{-1}\operatorname{sym}\nabla p^\flat,\qquad
 x=sw,\qquad h=x\bar g,\qquad A=Ns\bar g^{-1}.
 \end{gathered}
\end{equation}
The first relation uniquely defines \(s>0\), smoothly also at \(df=0\).
The vector \(p\) in this display is not a component of the mass
covector.  In particular
\(\bar g^{-1}=g^{-1}-v\otimes v\) and
\(\dd V_{\bar g}=W\,\dd V_g\).
We solve
\begin{equation}\label{eq:ah-system}
 \divg_g p=0,\qquad B=\nabla N-Kp,\qquad
 \divg_g B=3N+\Sigma,\qquad
 -\divg_g(A\,dw)=3N+\Sigma x+P ,
\end{equation}
where \(\Sigma,P\ge0\).  At the inner boundary, whose normal
\(\nu\) points into \(\Omega\), prescribe
\begin{equation}\label{eq:ah-inner-data}
 f=tM,\qquad B\cdot\nu=-t b_S,\qquad (A\,dw)\cdot\nu=0 .
\end{equation}
Here \(0\le t\le1\) is a continuation parameter.  On the outer
sphere \(S_R=\{r=R\}\), with outward \(g\)-normal \(\nu_R\), prescribe
\begin{equation}\label{eq:ah-outer-data}
 f=0,\qquad B\cdot\nu_R=\partial_{\nu_R}V_0,\qquad
 w=V_0^{-1}.
\end{equation}
We keep \(K\) fixed throughout the continuation; this preserves
\(H_g(S)\le k\), where \(k=K(\nu,\nu)\).

The identities of Lemmas~\ref{lem:graph-curvature},
\ref{lem:graph-current}, and~\ref{lem:graph-boundary} apply with
\(\tr_gK=\tr_g\pi=0\).  Writing \(Q=B+A\,dw\), they give
\begin{align}
 \divg_g Q&=-(x-1)\Sigma-P,\nonumber\\
 Q&=N\bar\nabla x+x(\pi-K)p-(x-1)B,\label{eq:ah-current}\\
 x^2(\Scal_h+6)
 &=x\{D+|\pi-K|_g^2\}
   +\frac{3}{2x}|dx|_{\bar g}^2+6(x-1)^2+\frac{2P}{N},
 \label{eq:ah-scalar}\\
 D&=16\pi(\mu-J\cdot v)\ge0,\nonumber\\
 \left(\Delta_{\bar g}-3+
  N^{-1}\divg_g((\pi-K)p)\right)x&=-3-P/N .
 \label{eq:ah-x-equation}
\end{align}
The dominant energy condition and \(|v|<1\) give the displayed
sign of \(D\).  At \(S\), the height maximum principle will give
\(v=-z\nu\), \(0\le z<1\), and the boundary formula is
\begin{equation}\label{eq:ah-boundary-curvature}
 s\sqrt{x}\,H_h=N(H_g-zk)-t b_S .
\end{equation}
Consequently a suitable small inner flux will make the obstacle
boundary strictly mean concave wherever an enclosing minimizer
could touch it.

Choose a fixed \(p_1>2\) and an area tolerance \(0<\delta<1/4\).
For a large finite height \(M\), set
\begin{equation}\label{eq:ah-parameter-scales}
 R_s=M^{3/10},\qquad T=M^{4/5},\qquad L=M^2.
\end{equation}
The allowable smooth inner data form the convex class
\begin{equation}\label{eq:ah-flux-class}
 \mathcal B_M=
 \left\{b_S\in C^\infty(S):
 0\le b_S\le B_*,\
 \|b_S\|_{L^{p_1}(S,g)}\le M^{-1/(10p_1)}\right\}.
\end{equation}
The fixed constant \(B_*\) will be chosen using an upper lapse
estimate which depends only on the norm constraint in
\eqref{eq:ah-flux-class}, not on \(B_*\).

Let \(\epsilon>0\) be chosen after \(M,L\).  The unscaled volume
source is a product of smooth cutoffs, between \(0\) and \(L\),
which vanishes when \(r\ge2R_s\), \(x\ge1-\delta/2\), or
\(N\le\epsilon\), and equals \(L\) when
\[
 r\le R_s,\qquad x\le1-\delta,\qquad N\ge2\epsilon .
\]
The smooth cutoffs are defined also for negative arguments.
Two further sources, each bounded by a finite strength \(d_*\),
have the following support and full-strength regions:
\begin{equation}\label{eq:ah-penalty-cutoffs}
 \begin{array}{c|c|c}
 &\text{vanishes if}&\text{equals }d_*\text{ if}\\ \hline
 P_1& f\le M/2\ \text{or }w\ge4T&
       f\ge3M/4,\quad w\le2T\\
 P_2& N\ge4\epsilon\ \text{or }w\ge4T_2&
       N\le3\epsilon,\quad w\le2T_2 .
 \end{array}
\end{equation}
Here \(T_2=C_2(M,L)/\epsilon\), with \(C_2\) fixed below before
\(\epsilon\) is chosen.  During the continuation, multiply both
the volume source and \(P_1+P_2\) by \(t\); we continue to denote
the scaled sources by \(\Sigma\), \(P_1\), \(P_2\),
and \(P=P_1+P_2\).
All strengths and heights remain finite when solving the equations.
The limit \(R\to\infty\) is always taken with these internal
parameters fixed.

\subsection{Lapse estimates with only bounded drift}

\begin{lemma}\label{lem:ah-lapse}
For smooth solutions on sufficiently large truncations, uniformly
over \(0\le t\le1\) and \(b_S\in\mathcal B_M\),
\begin{equation}\label{eq:ah-lapse-bounds}
 N\le Cr,\qquad N\ge c(M)\epsilon r,\qquad
 \frac{|N-V_0|}{r}\le
 C(R_s/r)^{1+\kappa}\quad(r\ge2R_s),
\end{equation}
where \(0<\kappa<\tau-1\) is fixed and small.  The upper constant
uses only the \(L^{p_1}\) bound of \(b_S\).  On compact sets,
\(N\) has uniform Hölder and \(W^{1,q}\) bounds for some \(q>3\),
depending on \(M,L\), but independent of \(\epsilon,d_*\) and
derivatives of \(b_S\).

The positivity and lower bound remain true at a fixed point in
which the drift \(Z=K(v)\) is computed using a smooth positive
extension of the constitutive lapse argument, provided that the
actual lapse equation is
\(\divg_g(\nabla N-ZN)=3N+\Sigma\) and the cutoff
\(N\le\epsilon\) uses the actual lapse.
\end{lemma}

\begin{proof}
Only \(|Z|\le |K|\) is used.  First consider the density equation
\begin{equation}\label{eq:ah-homogeneous-lapse}
 \divg_g(\nabla H_1-ZH_1)=3H_1
\end{equation}
with the prescribed outer flux and inner outward flux \(t b_S\).
Its adjoint is \(\Delta_g+Z\cdot\nabla-3\), with ordinary
homogeneous Neumann boundary condition.  The maximum and boundary
point principles give zero adjoint kernel.  Scalar elliptic
Fredholm theory, deforming the lower terms to those of
\(\Delta_g-3\), gives index zero and hence unique solvability of
the primal problem.  Green's identity with the adjoint solution
of a nonnegative forcing, which is nonpositive, proves
\(H_1\ge0\): both prescribed outward fluxes are nonnegative, and
the outer one is strictly positive.  Local Harnack then gives
strict positivity.

We give the weighted estimate used here, including its uniformity
in the outer radius.  For \(-\Delta_g+3\), scalar and divergence
forcing of size \(O(r^{-j})\), \(0<j<3\), with natural flux data
in \(L^{p_1}(S)\) and of size \(O(R^{-j})\) on \(S_R\), yield
an \(O(r^{-j})\) solution, with a constant independent of \(R\).
The natural datum includes the flux of the divergence forcing.
To see this, construct a positive strict supersolution comparable
to \(r^{-j}\), constant on a fixed inner region.  Change its
logarithmic slope slowly from \(0\) to \(-j\) in a long interval
of \(\log r\) lying sufficiently far out.  The limiting radial
indicial slopes are \(-3\) and \(1\), so this retains
\((-\Delta_g+3)\varphi\ge c\varphi\).  Add a fixed sufficiently
large multiple of
\(R^{-j-a}r^a\), similarly continued as a constant inside, where
\(0<a<1\), to make its outer normal derivative nonnegative.
The resulting barrier stays comparable to \(r^{-j}\) on \(r\le R\).

For divergence forcing the barrier argument is justified by
normalization and compactness.  If the asserted bound fails,
divide a violating solution by the maximum of its absolute
value relative to the barrier, and normalize locally by the
barrier's size at a maximizing point.  The normalized forcing
and boundary data tend to zero.  Local Laplace \(W^{1,q}\)
estimates are available with
\(3<q<3p_1/2\): the boundary \(L^{p_1}\) datum belongs to the
required negative trace space by the two-dimensional boundary
Sobolev embedding.  These estimates give local uniform compactness.
The outer sphere charts have uniformly controlled geometry.
They may be formed by flowing
\(\nabla\eta/|d\eta|^2\), \(\eta=\operatorname{arsinh}r\);
the normal direction is orthogonal and the reflected principal
metric is uniformly Lipschitz.  Thus no high derivatives of
geodesic normal coordinates are needed.
At points escaping to infinity, subsequences of the normalized
geometries converge to hyperbolic geometry, the relative radius
to an exponential horospherical coordinate, and a nearby outer
boundary to a horosphere.  The strict supersolution and its
Neumann sign persist in these limits.  A nonzero maximum of the
limiting solution divided by that supersolution contradicts the
maximum or boundary point principle.  The same argument in a
fixed chart handles bounded maximizing points.  This proves the
claimed estimate.

Apply it to \(H_1-V_0\), retaining the drift as divergence forcing.
The background errors are \(O(r^{1-\tau})\), and
\(Z=O(r^{-\tau})\).  It follows that
\begin{equation}\label{eq:ah-homogeneous-asymptotic}
 |H_1-V_0|\le Cr^{-\kappa}.
\end{equation}
Here and below \(V_0\) is extended smoothly across the compact
part.  There is one additional compactness point in this
application: a failure with bounded maximizing points could
produce a nonzero limit \(H_\circ\ge0\), with bounded measurable
limit drift, zero natural flux, and
\(H_\circ=O(r^{-\kappa})\), solving
\eqref{eq:ah-homogeneous-lapse}.  The sign follows from positivity
of \(H_1\) before subtracting \(V_0\) and dividing by an unbounded
normalizing constant.  Regard \(ZH_\circ\) as divergence forcing
for \(-\Delta_g+3\) and apply the preceding estimate repeatedly.
This improves the decay to \(O(r^{-j'})\) for some \(2<j'<3\).
To justify each improvement on the full exterior, solve the
baseline problem on truncations with zero natural outer flux
and pass to a limit; its difference from \(H_\circ\) decays in
the previous weight and vanishes by the maximum principle.
The improved decay is integrable.  Testing the density equation
with radial cutoffs, equal to one near the finite boundary,
and moving the Laplacian onto the cutoffs gives
\(3\int_\Omega H_\circ=0\).  All shell terms vanish since
\(j'>2\).  This is impossible.  On bounded limiting truncations
the same contradiction follows by integrating the zero-flux
equation directly.  This proves
\eqref{eq:ah-homogeneous-asymptotic} with the asserted uniformity.
For \(b_S=0\), Harnack, also by homogeneous-flux reflection at
\(S\), and connectedness now give \(H_1\ge cr\).

The dual maximum principle compares \(N\) to the homogeneous
solution with the same drift and boundary data, giving
\(N\le H_1\le Cr\).  For the lower bound divide \(N\) by the
homogeneous solution with the same drift and \(b_S=0\).
If this ratio is below \(c\epsilon/R_s\), with \(c\) sufficiently
small, then \(N\le\epsilon\) wherever the spatial cutoff permits
\(\Sigma\ne0\).  Hence \(\Sigma=0\) on this sublevel.  The ratio
there solves an elliptic drift equation with no zeroth term.
Its outward derivative is nonnegative at the inner face and
strictly positive at the outer face when the ratio is below
the threshold.  The minimum and boundary point principles
therefore exclude such a minimum.  This proves the positive
floor, and also excludes negative lapses when constitutive
arguments have been extended.

Finally, on \(2R_s\le r\le R\), subtract \(V_0\) and treat \(ZN\)
as divergence forcing of size \(O(r^{1-\tau})\).
The inner Dirichlet values have size \(O(R_s)\), and the natural
outer error has the corresponding decaying size.  The weighted
argument above, now with the inner Dirichlet face, gives
\[
 |N-V_0|\le C R_s^{1+\kappa}r^{-\kappa}.
\]
The homogeneous contribution of the inner data is bounded by
the same barriers.  In the normalized argument the zero
Dirichlet trace also excludes a limit maximum at that face.
This proves the final estimate in \eqref{eq:ah-lapse-bounds}.
On fixed patches write the lapse equation as a metric Laplace
equation with bounded divergence forcing \(ZN\), bounded scalar
forcing \(3N+\Sigma\), and the prescribed \(L^{p_1}\) boundary
data.  The local \(W^{1,q}\) and Hölder estimates from
Proposition~\ref{prop:elliptic-inputs} prove the last assertion.
\end{proof}

\subsection{Finite slopes and uniform estimates at infinity}

\begin{lemma}\label{lem:ah-graph-estimates}
The solutions satisfy \(0\le f\le tM\).  On every fixed compact
subset, including \(S\),
\begin{equation}\label{eq:ah-s-floor}
 s\ge c_{\mathrm{loc}}(M,L)\epsilon ,
\end{equation}
and their height gradients have finite bounds for fixed
\(M,L,\epsilon\), independent of \(d_*\), \(R\), and derivatives
of \(b_S\).  Moreover,
\begin{align}
 \int_{\{f\in I\}}p\cdot df&\le C|I|,\qquad
 \int_{\Omega_R}(W^2-1)\,\dd V_g\le CM,\label{eq:ah-height-energy}\\
 |f|&\le C R_s^{-1}(R_s/r)^{4-\xi},\nonumber\\
 |v|+|\pi|+|\nabla\pi|&\le C(R_s/r)^{3-\xi}
       \quad(r\ge C_0R_s),\label{eq:ah-end-graph}
\end{align}
where \(I\) is any height interval, \(\xi>0\) can be fixed
sufficiently small, and the constants in
\eqref{eq:ah-height-energy}--\eqref{eq:ah-end-graph} are
independent of \(M,L,\epsilon,d_*\) for large \(M\).
The end estimates include the corresponding local Hölder
control and derivatives of \(v\) through order two.
\end{lemma}

\begin{proof}
The maximum principle for \(\divg_g p=0\) gives the height bound.
We spell out how the differential estimate in
Lemma~\ref{lem:gradient-maximum} supplies all the finite bounds
used here.  At nonzero gradient write
\[
 z=|v|,\quad y=z^2,\quad m_f=|df|,\quad
 q=\frac{1-y}{1+y},\quad
 \beta=1+q,\quad
 \mathcal P=g^{-1}+(q-1)e\otimes e,\quad e=\nabla f/m_f .
\]
The height equation is
\(\mathcal P:\nabla^2 f+\beta\,d\log N\cdot df=0\).
At an interior maximum of
\(G+j(f)+d_0\), where \(G=-\log s\), \(j'<0\),
\(j''>0\), and \(y\ge1/2\), that lemma gives
\begin{equation}\label{eq:ah-gradient-maximum}
 j''q m_f^2\le
 C\left[1+\frac{\Sigma}{N}
 +(1-y)^2(j'm_f)^2+|dd_0|^2+|\nabla^2d_0|\right].
\end{equation}
Its hypotheses hold because the lapse equation has precisely
the drift \(K(v)\), the tensor \(K\) and its divergence are
bounded on the relevant patches, and \(\Sigma\ge0\).
The cancellation of the Hessian of \(\log N\) in that lemma
requires no prior second-derivative estimate for \(N\).
Multiplying \eqref{eq:ah-gradient-maximum} by \(N^2\) controls
\((1-y)^{-1}\), using
\(Nm_f=z/(1-y)\) and the upper lapse bound; a positive lower
lapse is not needed in this step.

At either constant-height face let \(n\) point towards increasing
end direction.  When \(df\ne0\), \(e=-n\), and the boundary part
of the same lemma gives
\[
 G_n=-\frac{yH_g+\partial_n\log N}{1-y}.
\]
At \(S\), \(\partial_\nu\log N=-zk-tb_S/N\), so
\(G_\nu\ge zk/(1+z)\), a uniform lower bound.
Since \(j'f_\nu=-j'm_f>0\), a maximum at this face bounds \(m_f\).
At \(S_R\), \(H_g\) is bounded below by a positive constant,
\(K=o(1)\), and
\(N\partial_{\nu_R}\log N=K(p,\nu_R)+\partial_{\nu_R}V_0\).
Taking \(|j'|\) sufficiently small makes the outward derivative
of \(G+j(f)\) negative at large \(y\), bounded above by
\(-c/(1-y)\).  Thus an outer maximum is excluded, or its slope
is bounded by a localization derivative.  A sufficiently small
decreasing exponential \(j\) has both required derivative signs.

On a compact truncation these arguments give a finite global
slope bound.  To make it independent of \(R\) locally, set
\(d_0=k_0\log\chi\), \(k_0\ge2\), where \(\chi\) is a smooth
cutoff on a slightly larger compact region, constant normally
near \(S\) if that face is present.  Equation
\eqref{eq:ah-gradient-maximum} bounds \(W\chi\) at an interior
maximum; the preceding face argument does the same at \(S\).
When \(y<1/2\) this bound is immediate.  Applying the lapse floor
to the weighted maximum gives \eqref{eq:ah-s-floor}.

Since \(|p|\le N\), the flux through \(S\) is bounded independently
of all internal large parameters.  Test \(\divg_g p=0\) with
a clipped height whose derivative is one on \(I\) and zero
elsewhere.  Its boundary oscillation is at most \(|I|\), giving
the first estimate in \eqref{eq:ah-height-energy}.
The identity \(p\cdot df=W^2-1\), tested with the entire height,
gives the second one.

We next prove the initial uniform end bound \(f\le C/r\);
it cannot be obtained from a compact-region bound depending
arbitrarily on \(M\).  By Lemma~\ref{lem:ah-lapse}, \(N\asymp r\)
when \(r\ge CR_s\), for a fixed large \(C\).
Fix a radial segment of \(\eta\)-length at most one starting
at radius \(r_*\) there, and set \(F=r_*f\).  In a bounded
number of uniformly controlled metric charts covering its
neighborhood, consider the subgraph of \(F\) in the product
with a Euclidean line.  The product vector
\[
 \mathcal Z=(-p/r_*,1)
\]
is bounded and divergence free.  Its pairing with the upward
graph normal is
\[
 \frac{1+p\cdot df}{\sqrt{1+r_*^2|df|^2}}
 =\frac{W^2}{\sqrt{1+r_*^2|df|^2}}\ge c>0,
\]
using \(N/r_*\asymp1\) and the constitutive relation.
The divergence theorem therefore makes the subgraph locally
quasiminimizing for perimeter, with a uniform constant.
Filling or emptying a product ball bounds the graph perimeter
there by a constant times the trace area of either phase.
The four-dimensional isoperimetric inequality then gives,
for the positive volume \(V(h)\) of either phase in a ball,
\(V'(h)\ge cV(h)^{3/4}\).
Integration yields uniform two-sided volume density, and
relative isoperimetry yields a fixed positive graph-area
lower bound in each sufficiently small ball centered on
the graph.

If the height span of \(F\) on the radial segment is \(D\),
choose graph points with separated heights and disjoint
fixed-size product balls.  The preceding estimate bounds
the graph area in the surrounding tube and relevant height
band below by \(cD-C\).  Conversely, the constitutive relation
gives
\(\sqrt{1+r_*^2|df|^2}\le C(1+p\cdot df)\).
The base tube has bounded volume and the relevant interval
of \(f\) has length at most \(C(D+1)/r_*\).
Equation~\eqref{eq:ah-height-energy} bounds the graph area
above by \(C+C(D+1)/r_*\).  For large fixed starting radius
this bounds \(D\) uniformly.  If the segment meets \(S_R\),
oddly reflect \(f\), evenly reflect \(N\), and reflect the
metric in the orthogonal charts described in the lapse proof.
The zero trace of \(f\) makes the divergence equation valid
weakly after reflection.  The reflected height band adds
only an interval of comparable length to the energy estimate.
Thus the same estimate holds there.  Summing the bounds
\(C/r_*\) over consecutive outward unit segments, up to
the zero height at \(S_R\), proves \(f\le C/r\).

On far-out unit balls rescale \((f,N,p,s)\) by
\((r_*,r_*^{-1},r_*^{-1},r_*^{-1})\), respectively.
The equations there have \(\Sigma=0\) and retain their form.
The localized gradient estimate now gives uniform bounds
for the rescaled height gradient and uniform ellipticity.
The outer-face argument applies with its rescaled positive
flux datum.  The rescaled lapse has uniform \(W^{1,q}\) bounds.
Differentiating the divergence height equation, or using
difference quotients, gives scalar uniformly elliptic
divergence equations for its first derivatives, with divergence
forcing in \(L^q\), \(q>3\).  The inhomogeneous Hölder estimate
therefore bounds \(df\) in a Hölder norm, also at a
constant-height boundary by odd reflection.  The lapse
divergence equation then gives \(C^{1,\gamma}\) control,
followed by the nondivergence height equation and successive
Schauder estimates.  These steps use only the stated
\(C^{2,\alpha}\) metric and \(C^{1,\alpha}\) tensor bounds.
They also prove compact-set gradient Hölder estimates away
from infinity without differentiating \(b_S\).

On each far-out ball, the energy estimate for a height interval
of length \(C/r_*\), together with this Hölder control, implies
\(z=o(1)\) uniformly as \(r_*\to\infty\).
The derivative version of \eqref{eq:ah-lapse-bounds} implies
that \(\nabla\log N\) tends to the hyperbolic radial direction
as \(r/R_s\to\infty\).  The limiting height operator is
\(\Delta_b+2\,d\eta\cdot d\); its decaying radial indicial
root is \(-4\).  For any sufficiently small fixed \(\xi>0\),
a positive multiple of \(r^{-4+\xi}\) is therefore a strict
supersolution sufficiently far beyond \(R_s\).
Comparison with the already proved \(C/r\) bound at a fixed
multiple of \(R_s\) gives the first estimate in
\eqref{eq:ah-end-graph}.  Treating the actual height equation
as a linear equation on these smaller height ranges gives
the corresponding derivative bounds by Schauder theory.
Inserting them in \(p=s^2\nabla f\) and
\(\pi=N^{-1}\operatorname{sym}\nabla p^\flat\) gives the
remaining assertions.
\end{proof}

For a fixed sufficiently large \(C_0\), all sources vanish when
\(r\ge C_0R_s\).  Indeed the height is then below \(M/2\),
the lapse exceeds \(4\epsilon\), and the spatial cutoff
for \(\Sigma\) has ended.  In this region
\begin{equation}\label{eq:ah-end-x}
 (\Delta_{\bar g}-3-U_0)x=-3,\qquad
 U_0=-N^{-1}\divg_g((\pi-K)p),\qquad
 |U_0|_{C^{0,\gamma}_{\mathrm{loc}}}
       \le C(R_s/r)^\lambda
\end{equation}
for some \(\lambda>3\).  For example one can take any exponent
below \(\min(6-2\xi,\tau+3-\xi)\), by
\eqref{eq:ah-end-graph}.  The constants here retain the
independence asserted in that lemma.

\subsection{Costs and enforcement of the finite penalties}

\begin{lemma}\label{lem:ah-penalties}
The parameters can be chosen in the order already specified,
with \(\epsilon\) sufficiently small after \(M,L\), and then
\(d_*\) sufficiently large but finite, such that
\begin{equation}\label{eq:ah-penalty-costs}
 \int_{\Omega_R}P_1\,\dd V_g\le C T/M,\qquad
 \int_{\Omega_R}P_2\,\dd V_g=o_{M,L}(1)
       \quad(\epsilon\downarrow0),
\end{equation}
uniformly in \(R,b_S\), and in the strengths.
At \(t=1\), the same choices enforce
\begin{equation}\label{eq:ah-penalty-targets}
 w|_S\ge T,\qquad
 w\ge T_2\quad\hbox{on }\{N\le2\epsilon\},\qquad
 sT_2\ge1\quad\hbox{on }\{N\le4\epsilon\}.
\end{equation}
Consequently \(x<1-\delta\) and \(r\le R_s\) imply
\(\Sigma=L\).
\end{lemma}

\begin{proof}
The lapse floor confines \(N<6\epsilon\) to a compact set
depending on \(M\), independently of \(\epsilon\).
Equation~\eqref{eq:ah-s-floor} on a slightly larger compact set
therefore permits a choice of \(C_2(M,L)\) such that
\(sT_2\ge1\) throughout the required low-lapse region.

For every Lipschitz test \(\zeta\) vanishing at \(S_R\), positivity
of \(w\), its equation, and its zero inner flux give
\begin{equation}\label{eq:ah-penalty-test}
 \int P\,\frac{\zeta^2}{w}\,\dd V_g
       \le \int A(d\zeta,d\zeta)\,\dd V_g .
\end{equation}
Indeed test with \(\zeta^2/w\) and complete the square in
\(2\zeta\,dw\cdot A\,d\zeta/w-\zeta^2 A(dw,dw)/w^2\).
All the other terms on the right side of the \(w\)-equation
are nonnegative.  On the support of \(P_1\),
\(f/M\ge1/2\) and \(w\le4T\).  With \(\zeta=f/M\),
the algebraic inequality
\(A(df,df)=W-W^{-1}\le W^2-1=p\cdot df\) and
\eqref{eq:ah-height-energy} prove
\(\int P_1\le CT/M\).

For \(P_2\) take a cutoff of \(N/\epsilon\), equal to one
below \(4\) and zero above \(6\).  It vanishes near \(S_R\)
for large \(R\).  Since \(A\le N^2g^{-1}\) and \(w\le4T_2\)
on this source's support,
\begin{equation}\label{eq:ah-low-lapse-cost}
 \int P_2\le
 C T_2\int_{\{N<6\epsilon\}}|\nabla N|^2\,\dd V_g.
\end{equation}
Test the lapse equation with \((6\epsilon-N)^+\).
The outward current flux at \(S\) is \(t b_S\ge0\), so its
boundary contribution has the favorable sign.  On the
sublevel set \(|Kp|\le CN\le C\epsilon\).  Cauchy's inequality
then gives
\begin{equation}\label{eq:ah-low-lapse-energy}
 \int_{\{N<6\epsilon\}}|\nabla N|^2
 \le C(M)\epsilon^2
       +C\epsilon\int_{\{N<6\epsilon\}}\Sigma .
\end{equation}

We must show that the final integral tends uniformly to zero;
boundedness of \(\Sigma\) alone would not suffice.
For any potentially violating sequence, use the uniform
local lapse Hölder bounds to take \(N_j\to N_0\ge0\)
uniformly on the compact region containing these sublevels.
After a subsequence,
\[
 Kp_j\rightharpoonup Z_0N_0,\qquad
 \Sigma_j\rightharpoonup^*\Sigma_0,\qquad
 |Z_0|\le C,\quad 0\le\Sigma_0\le L .
\]
The weak limit equation is
\(\Delta_gN_0=\divg_g(Z_0N_0)+3N_0+\Sigma_0\).
We claim \(\Sigma_0=0\) almost everywhere on \(\{N_0=0\}\).
Choose an interior density point of that zero set which is
also a Lebesgue point of \(\Sigma_0\).  In balls of radius
\(h\downarrow0\), rescale \(N_0\) by \(h^{-2}\).
The rescaled fields are nonnegative and vanish at the center.
Their divergence drift has size \(O(h)\) and their scalar
forcing is bounded.  The inhomogeneous Harnack estimate
therefore gives a uniform bound on smaller balls.
Because the rescaled zero sets have asymptotically full
measure, these functions tend to zero in local \(L^1\).
Testing the rescaled equation against compactly supported
smooth functions shows that the limiting constant scalar
forcing is zero, proving the claim.  Boundary points need
not be considered, since \(S\) has zero volume.

For completeness, the changing sublevel sets cause no loss:
for every \(a>0\), uniform convergence gives
\(\{N_j<6\epsilon_j\}\subset\{N_0\le a\}\) eventually.
Nonnegativity and weak convergence, using continuous cutoffs
of \(N_0\) that majorize this fixed sublevel, imply
\[
 \limsup_j\int_{\{N_j<6\epsilon_j\}}\Sigma_j
 \le \int_{\{N_0\le2a\}}\Sigma_0.
\]
Letting \(a\downarrow0\) proves the asserted uniform convergence.
Equations~\eqref{eq:ah-low-lapse-cost} and
\eqref{eq:ah-low-lapse-energy}, with \(T_2=C_2/\epsilon\),
prove the second cost in \eqref{eq:ah-penalty-costs}.
Choose \(\epsilon=\epsilon(M)\) sufficiently small that this
cost, as well as \(T/M=M^{-1/5}\), tends to zero as \(M\to\infty\).

Now fix \(M,L,\epsilon\) and increase the finite strengths.
The previously proved continuity of \(f,N\) supplies uniform
neighborhoods of the targets \(S\) and \(N\le2\epsilon\)
inside, respectively, the weaker triggering regions
\(f\ge3M/4\) and \(N\le3\epsilon\).
The cost bounds imply that the volume in these neighborhoods
where \(w\le2T\), or \(w\le2T_2\), tends to zero as
\(d_*\to\infty\).
For either threshold \(D=T,T_2\), the bounded nonnegative
function \((2D-w)^+\) is a weak subsolution of the homogeneous
divergence operator \(\divg_g(A\,d\cdot)\).
Its coefficients are uniformly elliptic on the relevant
compact region for these fixed parameters, independently of
the strength.  The scalar local supremum estimate bounds its
supremum on a smaller neighborhood by its \(L^2\) mean on the
larger one.  At \(S\), even reflection using the zero natural
flux gives the same estimate.  Choosing \(d_*\) sufficiently
large makes that supremum smaller than \(D\).
This proves \eqref{eq:ah-penalty-targets}, with room in the
thresholds if desired.  Finally, if \(x<1-\delta\), the
second target excludes \(N\le2\epsilon\), so the volume
penalty is at full strength when also \(r\le R_s\).
\end{proof}

\subsection{Compactness, mass, and a uniform comparison sphere}

Until the existence argument below, a \emph{limit solution} means
any limit along expanding truncations of smooth solutions at
the fixed internal parameters and fixed \(t\), allowing arbitrary
\(b_S\in\mathcal B_M\).  Statements about this family are a priori
statements and do not presuppose that it is nonempty.
Pass also to a weak \(L^{p_1}(S)\) limit of the boundary data.
Until the selector is imposed, we use \(b_S\) for this inherited
datum in a limit solution: it need not be smooth, but its
pointwise bounds, norm bound, and natural-flux identity pass
to the limit.  Its integral is the limit of the integrals of
the smooth data.

\begin{lemma}\label{lem:ah-limit-mass}
With internal parameters fixed, the fields \(N,f,df,w\)
are uniformly precompact in local uniform topology, including
at \(S\).  Limit solutions have \(N,w>0\) and are classical
away from \(S\).  They obey
\begin{equation}\label{eq:ah-conformal-decay}
 x-1=O_{\mathrm{fixed},2,\gamma}(r^{-3+\xi}).
\end{equation}
Here \(\xi\) is taken sufficiently small, in particular
\(0<\xi<\min(3-\tau,\tau-1)\).
Their metrics \(h\) satisfy the required asymptotically
hyperbolic decay, all four mass fluxes are finite, and
\(V_0(\Scal_h+6)\) is integrable near infinity.  Their time
components satisfy
\begin{equation}\label{eq:ah-mass-budget}
 8\pi\{p_0(h)-p_0(g)\}
   =t\int_S b_S\,\dd A_g
      +\int_\Omega\{(x-1)\Sigma+P\}\,\dd V_g
   \le o(1)\quad(M\to\infty).
\end{equation}
The \(o(1)\) is uniform over the allowed data, with the
subsequent choices of \(\epsilon,d_*\) just described.
\end{lemma}

\begin{proof}
Only the upper bound for \(w\) remains to be established.
The right side of its equation is bounded on compact sets:
\(\Sigma x\le L\), since a nonzero \(\Sigma\) requires \(x<1\),
and \(P\) has fixed finite strength.
By \eqref{eq:ah-lapse-bounds} and \eqref{eq:ah-end-graph},
a multiple of \(r^{-1}\) is a supersolution for
\(-\divg_g(A\,dw)=3N\) sufficiently far beyond \(R_s\).
Comparison therefore bounds the end values of \(w\) by
\[
 C\left(1+\sup_{\{r\le C_0R_s\}}w\right)r^{-1}.
\]
If its compact supremum were unbounded, divide by that
supremum.  Uniform ellipticity, local divergence estimates,
and convergence of \(N,df\) then give a nonzero nonnegative
limit satisfying the homogeneous \(w\)-equation, with zero
inner flux and decaying to zero at infinity.  The maximum
principle excludes such a function; Harnack and reflection
at \(S\) handle a maximum at that face.  Thus the compact
upper bounds hold.  Local Hölder estimates for \(w\) and
the already proved estimates for \(N,df\) give the stated
precompactness.  The lapse floor and the strictly positive
forcing \(3N\) in the supersolution equation imply positivity
of both limiting fields.  Scalar elliptic bootstrapping gives
classical interior regularity.

On the end, \eqref{eq:ah-end-x} and these upper bounds permit
comparison for \(x-1\) with a multiple of \(r^{-3+\xi}\).
For a finite truncation add \(C(r/R)^{a_0}\), \(0<a_0<1\),
to dominate its outer boundary values.  The first term
dominates the decaying forcing \(U_0\), while the second
vanishes on fixed sets as \(R\to\infty\).
The scalar zeroth coefficient is positive after increasing
the fixed end radius.  The maximum principle proves the
zeroth-order bound in \eqref{eq:ah-conformal-decay}, and
interior Schauder estimates prove its derivative and Hölder
versions.  These estimates are uniform over all limit
solutions at the fixed parameters, without bounds on
derivatives of \(b_S\).

Since
\(\bar g-g=O_{\mathrm{fixed},2,\gamma}(r^{-6+2\xi})\),
the choice \(\xi<3-\tau\) ensures that \(h-b\) has the
original admissible decay exponent, with a possibly smaller
Hölder exponent.  Put \(\varphi=x-1\).
The change in each defining mass flux is, up to an error
tending to zero, the flux of the one-form
\[
 -2\{V_\mu\,d\varphi-\varphi\,dV_\mu\}.
\]
All nonlinear conformal errors and graph terms disappear
by the displayed decay.  This flux has a finite limit:
its divergence is
\(-2V_\mu(\Delta_b-3)\varphi\), and
\eqref{eq:ah-end-x} gives absolute weighted integrability.
Indeed \(U_0=O(r^{-\lambda})\), \(\lambda>3\);
the difference between \(\Delta_{\bar g}\) and \(\Delta_b\)
on \(\varphi\) is
\(O(r^{-\tau-3+\xi})+O(r^{-9+3\xi})\).
Multiplication by \(V_0\) and integration against the
hyperbolic volume, of radial size \(r\,\dd r\), is integrable.
The scalar identity \eqref{eq:ah-scalar}, the original
weighted matter and \(K^2\) integrability, and
\eqref{eq:ah-end-graph}--\eqref{eq:ah-conformal-decay}
similarly prove integrability of \(V_0(\Scal_h+6)\).

On each limit end the lapse equation also gives
\[
 (\Delta_g-3)(N-V_0)
  =O(r^{1-\tau})+\divg_g(Kp).
\]
Together with \eqref{eq:ah-lapse-bounds} and the end
derivative estimates this yields
\(|\nabla(N-V_0)|=O_{\mathrm{fixed}}(r^{-\kappa})\).
Insert the second expression for \(Q\) in
\eqref{eq:ah-current}.  Its sphere flux differs from
that of \(V_0\,d\varphi-\varphi\,dV_0\) by a quantity
tending to zero.  The \(N-V_0\) terms have integrated
size \(O(r^{-1-\kappa+\xi})\); the
\((\pi-K)p\) terms have integrated sizes bounded by
\(O(r^{-3+2\xi})+O(r^{-\tau+\xi})\).
Metric replacement errors vanish by the same decay.
It follows that
\[
 \lim_{r\to\infty}\int_{\{r=\mathrm{const}\}}Q\cdot\nu_r
       \,\dd A_g=-8\pi\{p_0(h)-p_0(g)\}.
\]
At the inner boundary the outward normal of the exterior
is \(-\nu\), and \(Q\cdot(-\nu)=t b_S\).
Integrating \(\divg_g Q=-(x-1)\Sigma-P\) proves
\eqref{eq:ah-mass-budget}.  This integration remains valid
for limits with only continuous fields at \(S\): use the
first expression \(Q=A\,dw+B\), the weak equations and
their prescribed natural fluxes on compact regions.
The second expression is needed only at infinity, where
the fields are classical.  Finally,
\((x-1)\Sigma\le0\), the \(L^{p_1}\) bound makes
\(\int_S b_S=o(1)\), and
Lemma~\ref{lem:ah-penalties} gives \(\int P=o(1)\).
\end{proof}

The next estimate is needed before choosing \(b_S\); compact
upper bounds for \(w\) at fixed parameters alone would not
give it.

\begin{lemma}\label{lem:ah-comparison-sphere}
There is a fixed large \(C_1\) such that, at
\(r_1=C_1R_s\), every limit solution at \(t=1\) satisfies
\begin{equation}\label{eq:ah-comparison-area}
 \Area_h(\{r=r_1\})\le C R_s^2,
\end{equation}
with \(C\) independent of the internal large parameters.
\end{lemma}

\begin{proof}
On \(r>r_1\) solve
\[
 (\Delta_{\bar g}-3-U_0)Y=0,\qquad
 Y|_{r=r_1}=0,\qquad Y/V_0\longrightarrow1 .
\]
Existence follows by solving with outer Dirichlet datum
\(V_0\) on larger truncations, applying positive barriers,
and passing to a limit.  Choose
\(\xi<b_0<\min(1,\tau)\).
Positive and negative barriers of size
\(Cr(r_1/r)^{b_0}\) for \(Y-V_0\) work by the end estimates.
They give \(0\le Y\le Cr\), and the corresponding first
derivative control follows from elliptic estimates.
On an annulus from \(r_1\) to a large fixed multiple of
\(r_1\), compare from below with a small positive multiple
of \(r_1((r/r_1)^2-1)\).  This is a subsolution with zero
inner value, and its outer value can be made smaller than
that of \(Y\).  The boundary point principle and local
boundary estimates therefore give, in the end direction,
\[
 cr_1\le\partial_nY\le Cr_1
      \quad\hbox{on }\{r=r_1\},
\]
where \(n\) is the \(\bar g\)-unit normal.

For \(\varphi=x-1\), the operator just used satisfies
\((\Delta_{\bar g}-3-U_0)\varphi=U_0\).
Green's identity, the mass-flux calculation, and
\(b_0>\xi\) give
\begin{equation}\label{eq:ah-green-comparison}
 \int_{\{r=r_1\}}(x-1)\partial_nY\,\dd A_{\bar g}
 =\int_{\{r>r_1\}}YU_0\,\dd V_{\bar g}
       +8\pi\{p_0(h)-p_0(g)\}
 \le Cr_1^3 .
\end{equation}
The source integral is bounded by \(Cr_1^3\), since
\(\lambda>3\), \(Y\le Cr\), and \(W\) is uniformly bounded
on this end region.  The mass increment is bounded above
by \eqref{eq:ah-mass-budget}.  Add
\(\int\partial_nY\le Cr_1^3\) to both sides of
\eqref{eq:ah-green-comparison}, and then use \(x>0\) and
\(\partial_nY\ge cr_1\).  This yields
\(\int_{\{r=r_1\}}x\,\dd A_{\bar g}\le Cr_1^2\).
Two-dimensional conformal scaling makes this integral
exactly the area in \eqref{eq:ah-comparison-area}.
\end{proof}

\subsection{A continuous choice of inner flux}

\begin{lemma}\label{lem:ah-contact-selector}
For all sufficiently large \(M\), at the finite parameters
chosen above, there is a continuous rule \(\mathfrak b\) on
the uniform topology of the fields \((N,f,df,w)\) on a fixed
compact original subdomain, with values in a finite convex
hull in \(\mathcal B_M\), such that every limit solution at
\(t=1\) satisfies
\begin{equation}\label{eq:ah-contact-majorant}
 \mathfrak b(N,f,df,w)>N(H_g-zk)
\end{equation}
at every possible contact on \(S\) of a perimeter-minimizing
enclosure of \(S\).
\end{lemma}

\begin{proof}
Consider all limits at \(t=1\) arising from arbitrary smooth
choices \(b_S\in\mathcal B_M\) on arbitrary expanding
truncations.  Their restrictions to each fixed compact
region form a compact family in the uniform field topology.
Precompactness follows from Lemma~\ref{lem:ah-limit-mass};
closedness follows by a diagonal choice of actual truncation
solutions from the defining sequences of a convergent
sequence of limits.  This definition uses no existence
assertion, and the family may initially be empty.

The fixed-parameter end estimates are uniform over this
family.  Choose a large fixed radial sphere, beyond \(r_1\),
so that all spheres from slightly before it to infinity are
strictly outward mean convex in every limit metric.
Adjoin smooth compact orientable handlebodies to all components of
\(S\), possible because each is a closed orientable surface.  Choose
the gluing maps to reverse the induced boundary orientations, so the
orientation of \(\Omega\) extends over the caps.
Let \(O\) denote their union and \(X\) the resulting
one-ended smooth manifold without boundary; let \(K_c\)
be the compact region cut off by the chosen large sphere.
Extend each continuous positive metric through \(S\) to
the cap side, continuously with respect to its uniform
field values.  For example reflect in a fixed base collar
and interpolate to a fixed cap metric.  The mixed metric
entries vanish at \(S\), because \(f\) has constant trace.
Perimeters of sets containing \(O\) depend only on the
metric outside \(O\) and its boundary values.

For each of these metrics minimize perimeter among
finite-perimeter subsets of \(K_c\) containing \(O\)
up to null sets.  Compactness and lower semicontinuity
give a minimizer.  These facts hold for a continuous
positive metric by uniform smooth approximation of the
area integrands.  The sphere at \(r_1\) is an admissible
comparison, so every minimizer has perimeter at most
\(CR_s^2\).

Define contact points on \(S\) as points in the support
of a minimizer's perimeter measure.  The union of all
such contacts equals the contact set of one minimizer.
To prove this, minimize volume in an auxiliary fixed
smooth metric among the perimeter minimizers.  Perimeter
submodularity shows that the intersection and union of
any two minimizers are still minimizers.  The
volume-minimizing one, denoted \(E_-\), is therefore
contained almost everywhere in every other minimizer.
If a contact of a second minimizer were absent from the
perimeter support of \(E_-\), a small ball there would
have zero \(E_-\)-perimeter.  Since \(O\) occupies a
positive part of that ball, the ball would be filled
by \(E_-\), and hence by the second minimizer, a
contradiction.  The contact union is consequently the
closed contact set of \(E_-\).

These contact unions are upper semicontinuous in the
uniform field topology.  Indeed, along uniformly
convergent metrics, any sequence of minimizers has an
\(L^1\) subsequence converging to a minimizer for the
limit metric.  At a contact point, the complement has
uniform positive volume density: filling a chart ball
is an allowed competitor, so its perimeter in the ball
is bounded by a constant times its trace area on the
sphere.  Isoperimetry, applied to the complement inside
the ball, gives \(V^{2/3}\le CV'\), and thus
\(V(r)\ge cr^3\).  Constants are uniform on the compact
family of positive continuous metrics.  The cap supplies
positive density for the set on the other side.
These two density bounds persist when both the metric
and the contact point converge, so the limit point is
in the perimeter support of a limit minimizer.

The same density bounds place all contacts in the
measure-theoretic boundary.  Up to a set of
two-dimensional Hausdorff measure zero they therefore
belong to the reduced boundary, by the finite-perimeter
structure theorem.  At almost every point of their
intersection with the smooth surface \(S\), the tangents
are those of \(S\).  Since \(df\) is tangentially zero,
the area density there is \(x\,\dd A_g\).
Writing \(C_h\) for the contact union, we obtain
\begin{equation}\label{eq:ah-contact-area}
 \int_{C_h}x\,\dd A_g\le CR_s^2.
\end{equation}

Set \(\Theta=[N(H_g-zk)]_+\) on \(S\).
Since \(H_g\le k\), its positive part is bounded by \(Cs\).
For example, when \(k\ge0\),
\(N(H_g-zk)\le Nk(1-z)\le ks\); when \(k<0\)
the left side is nonpositive.  The upper lapse bound
also gives \(\Theta\le C\), with \(C\) independent of
the pointwise cap \(B_*\).  Fix \(B_*\) strictly above
this latter bound.  By \eqref{eq:ah-penalty-targets},
\(s\le x/T\) at \(S\), and hence
\begin{equation}\label{eq:ah-contact-flux-cost}
 \int_{C_h}\Theta^{p_1}\,\dd A_g
 \le C\int_{C_h}s\,\dd A_g
 \le C R_s^2/T=CM^{-1/5}.
\end{equation}
The admissible \(p_1\)-power budget in
\eqref{eq:ah-flux-class} is \(M^{-1/10}\).
Thus for sufficiently large \(M\) there is strict slack.
For each field in the compact family, approximate its
continuous nonnegative threshold on the closed set \(C_h\)
from above by a smooth function, with a small positive
margin, and taper it to zero in a sufficiently small
neighborhood of that set.  Regularity of surface measure
and \eqref{eq:ah-contact-flux-cost} keep its \(L^{p_1}\)
norm within the budget; the fixed gap below \(B_*\)
keeps its pointwise values admissible.  This constructs
a datum strictly majorizing the threshold on \(C_h\).
When \(C_h\) is empty the condition is vacuous.

Upper semicontinuity of contacts and continuity of the
threshold show that the same datum works in an open
neighborhood of the given field in the compact limit
family.  Take a finite such cover, with associated
smooth data \(b_1,\dots,b_m\), and enlarge the neighborhoods
to open subsets of the ambient uniform field space so
that their intersections with the limit family retain
this property.  A continuous partition of unity,
supplemented by the complement of the compact limit
family carrying datum zero, yields \(\mathfrak b\).
Explicitly, distance to the complements of these finitely
many open neighborhoods, together with distance to the
compact limit family, gives nonnegative continuous weights
with nonzero sum after a harmless subordinate refinement.
Normalize and take the corresponding convex combination.
At a field in the limit family the zero datum has weight
zero, and every active \(b_i\) is a strict majorant
at all its contacts.  Convexity preserves both constraints
in \eqref{eq:ah-flux-class}.  If the family is empty, use
the constant zero rule.  Only the original compact
subdomain's fields enter this rule; no trial fields on
the caps are required.
\end{proof}

\subsection{Existence at fixed finite parameters}

\begin{proposition}\label{prop:ah-finite-existence}
With the choices above, the system
\eqref{eq:ah-system}--\eqref{eq:ah-outer-data}, using
\(b_S=\mathfrak b(N,f,df,w)\), has a smooth positive
solution at \(t=1\) on every sufficiently large fixed
truncation.  Expanding truncations have a subsequence
converging to a solution smooth up to \(S\) on the
whole exterior.  It satisfies all the preceding estimates
and realizes the rule \(\mathfrak b\).
\end{proposition}

\begin{proof}
We describe the compact map precisely, because a frozen
source need not preserve positivity of a lapse output
away from fixed points.  Fix a small Hölder exponent
\(\gamma>0\), to be decreased below the exponents in
the a priori estimates, and use triples of
\(C^{1,\gamma}\) functions on the fixed truncation.
Choose \(a>0\) so that \(2a\) is below the fixed-point
floor in Lemma~\ref{lem:ah-lapse}, and a smooth function
\(\mathcal E:\R\to[a,\infty)\) equal to the identity
on \([2a,\infty)\).

Given trial fields and \(t\), first solve the linear
nondivergence height equation, freezing
\(\mathcal P,\beta,d\log\mathcal E(N)\) from the
trial arguments, with Dirichlet values \(tM\) and zero.
At zero trial gradient the smooth continuations are
\(\mathcal P=g^{-1}\) and \(\beta=2\).
The first-order coefficient acts on the new height.
Next compute \(v\) from this new height and the
extended trial lapse, and set \(Z=K(v)\).
Solve the linear density equation for the raw new lapse:
\[
 \divg_g(\nabla N_{\mathrm{new}}-ZN_{\mathrm{new}})
    =3N_{\mathrm{new}}+\Sigma_{\mathrm{trial}} .
\]
Use the prescribed outer current flux and inner current
flux \(-t\mathfrak b\) evaluated on the trial fields.
The source here is frozen from the trial fields,
including the raw trial lapse in its cutoff at
\(N\le\epsilon\).  The scalar multiplying the drift
and the zeroth term is the raw output lapse, not
\(\mathcal E(N_{\mathrm{new}})\).
The adjoint argument in Lemma~\ref{lem:ah-lapse}
proves unique linear solvability.

Finally solve a linear \(w\)-equation whose coefficient
\(A\) is formed from the new height and
\(\mathcal E(N_{\mathrm{new}})\).  Its right side is
\[
 3\mathcal E(N_{\mathrm{new}})
    +\Sigma_{\mathrm{trial}}x_{\mathrm{trial}}^+
    +P_{\mathrm{trial}},
 \qquad
 x_{\mathrm{trial}}^+
  =s(\mathcal E(N_{\mathrm{trial}}),df_{\mathrm{trial}})
      (w_{\mathrm{trial}})^+ .
\]
It has the zero inner natural flux and the prescribed
positive outer Dirichlet datum.  The three solves are
sequential, and define a continuous compact map
\(\mathcal T_t\).
Both frozen sources include their prescribed factor \(t\).
Indeed bounded sets of input fields give uniform
ellipticity and Hölder coefficients.  The height output
is bounded in \(C^{2,\gamma}\), so the drift in the
lapse step is \(C^{1,\gamma}\); its inverse estimates
are uniform on bounded input sequences by Schauder
theory, coefficient compactness in a lower exponent,
and the zero-kernel adjoint argument.  The lapse output
and then the \(w\) output have second-order Hölder
bounds.  The rule \(\mathfrak b\) has values in a
fixed finite convex hull of smooth functions, hence
uniform bounds of every spatial derivative.  The
compact embedding of these \(C^{2,\gamma}\) bounds
into \(C^{1,\gamma}\) proves compactness.  The same estimates and
uniqueness prove continuity without differentiating
the selection rule.

At a fixed point, the cutoff in the actual lapse
variable and the bounded drift give the positive floor
in Lemma~\ref{lem:ah-lapse}.  Thus every extension
\(\mathcal E\) becomes the identity.  The last equation
and positive outer datum give \(w>0\), so the positive
part also disappears.  Every fixed point therefore
satisfies the original equations and boundary data.

All such fixed points for \(0\le t\le1\) have bounded
\(C^{1,\gamma}\) norms for the fixed truncation and
fixed parameters.  Here is the regularity chain
underlying this assertion.  The lapse bound, height
gradient bound, and bounded sources give lapse
\(W^{1,q}\) estimates, \(q>3\), and Hölder estimates
for the height gradient, as in
Lemma~\ref{lem:ah-graph-estimates}.  The lapse divergence
equation then gives \(C^{1,\gamma'}\) estimates for
some \(\gamma'>0\); its scalar-forcing portion can
equivalently be estimated by the Laplace \(W^{2,q}\)
theorem.  The selected boundary data have bounded
smooth norms.  The nondivergence height equation
gives \(C^{2,\gamma'}\) estimates, after decreasing
\(\gamma'\) if necessary.  The \(w\)-equation has
bounded forcing, since \(\Sigma x\le L\) and \(P\)
has finite strength.  Mixed-boundary coercivity first
gives an \(H^1\) bound, then scalar local boundedness
and Hölder estimates give a uniform bound and
continuity.  Successive Schauder estimates now give
second-order Hölder bounds for all fields.  Take
the Banach exponent \(\gamma<\gamma'\).
The same bootstrap makes every fixed point smooth.

At \(t=0\), the map is constant.  The first output
is \(f_0=0\), so the drift in the second step is
zero independently of the input.  Its output is
the unique positive solution \(N_0\) of
\(\Delta_gN_0=3N_0\), with zero inner flux and the
prescribed outer flux.  The third output \(w_0\)
uses \(\mathcal E(N_0)=N_0\), by the homogeneous lapse
floor, and is the unique solution of
\[
 -\divg_g(N_0^2\nabla w_0)=3N_0,\qquad
 \partial_\nu w_0|_S=0,\qquad w_0|_{S_R}=1/V_0 .
\]
In general \(w_0\ne1/N_0\), because the outer lapse
condition is Neumann.  Choose a ball in the Banach
space containing this constant output and all
homotopy fixed points.  On its boundary
\(\Id-\mathcal T_t\) never vanishes, and its
Leray--Schauder degree is \(+1\) at \(t=0\).
Homotopy invariance gives a fixed point at \(t=1\).

Now let \(R\to\infty\).  The fixed-parameter compact
estimates give a subsequential limit on every compact
set, and a diagonal subsequence gives a global limit.
It belongs to the family used to define \(\mathfrak b\).
Uniform convergence of \((N,f,df,w)\) on its defining
compact domain implies convergence of the selected
datum, in every spatial norm because its range is
a finite smooth convex hull.  Thus the limit realizes
the rule, and the elliptic bootstrap is valid up to
\(S\) as well as in the interior.
\end{proof}

\subsection{A free minimal boundary}

Apply Proposition~\ref{prop:ah-finite-existence} at \(t=1\).
For this smooth metric, extend through the cap side
smoothly, and consider the perimeter minimization in
Lemma~\ref{lem:ah-contact-selector}.  Equation
\eqref{eq:ah-boundary-curvature} and the strict majorant
give \(H_h(S)<0\) at every possible contact.
This excludes contact without any assumed regularity
of the minimizing surface at the obstacle.

Here is the finite-perimeter argument.  Near the closed
contact set, the unit normals to sufficiently small
parallel outward collars of \(O\) have negative
divergence.  Off a neighborhood of that set, a thin
collar is already filled up to null sets: there is no
perimeter support on \(S\), and the cap side is filled.
Adjoin the entire collar of width \(a\) to a minimizer.
For almost every sufficiently small \(a\), Gauss--Green
and slicing bound the perimeter change above by the
integral of that negative divergence on the added part.
The comparison field has unit flux on the new outer
leaf, and its flux across the removed old boundary is
bounded by the area removed.  If contact existed,
the complement density proved above would give
positive added volume, so the perimeter would strictly
decrease.  This is impossible.

There is likewise no contact with the outer constraint
\(\partial K_c\).  The outward normals to the far
coordinate spheres have positive divergence, so
cutting away the outside part of a competitor decreases
perimeter whenever it removes positive volume.
Gauss--Green and slicing make this statement valid
for finite-perimeter competitors at almost every
cutting radius.  It also shows that no larger precompact
competitor, unconstrained by \(K_c\), can have smaller
perimeter: first trim it by such a sphere.

All minimizing boundaries are therefore free, locally
perimeter minimizing in a smooth three-dimensional
metric.  Interior codimension-one minimizing-boundary
regularity gives smooth compact embedded minimal
boundaries.  Choose a largest minimizer \(E\), by
maximizing auxiliary volume among the minimizers in
\(K_c\).  Its open representative contains \(O\)
compactly in its interior and is strictly outward
minimizing: equality for an enclosure differing by
positive volume would
contradict volume maximality after trimming.
Every component of \(E\) meets \(O\), since an
unattached component could be removed with a strict
perimeter saving.  Its open exterior is connected,
since filling any bounded exterior component would
also save perimeter.

Write
\begin{equation}\label{eq:ah-hull-area}
 A_h=\Area_h(\partial E)>0.
\end{equation}
The closed exterior \(X\setminus E\), with its boundary
included, is smooth, connected, complete, orientable, and one-ended.
Its orientation is the restriction of the orientation of \(X\);
the graph and conformal changes alter the metric, not this orientation.
Its compact truncations are compact, and its end is
asymptotically hyperbolic.  It satisfies
\(\Scal_h\ge-6\) by \eqref{eq:ah-scalar}, and its
weighted scalar integrability and finite mass fluxes
were verified in Lemma~\ref{lem:ah-limit-mass}.
Its minimum enclosing area is exactly \(A_h\).
Indeed any admissible end-side domain in this closed
exterior, on taking the complement of its interior
in \(X\), supplies a precompact finite-perimeter
enclosure of \(E\).  Its perimeter is bounded by the
entire intrinsic boundary area specified in the
definition, including any part coinciding with
\(\partial E\).  Outward minimization gives the
lower bound \(A_h\), and the original boundary
itself gives equality.

The time-symmetric hypothesis of
Theorem~\ref{thm:ah-reduction} is thus applicable to
this fixed metric and gives
\begin{equation}\label{eq:ah-ts-application}
 p_0(h)\ge F(A_h)\ge0.
\end{equation}
No causal property of the intermediate mass covector
has been used.

\subsection{The weighted bad-volume estimate and conclusion}

Suppose first that \(A_*>0\), fix \(0<\varepsilon_*<1\),
and choose the preceding \(\delta\) sufficiently small
for Proposition~\ref{prop:area-transfer}.
Fix a sufficiently large projection radius \(r_p\)
and put
\[
 \rho=\min(1,r_p^2/r^2),\qquad
 \mathcal D=\{x<1-\delta\}.
\]
By \eqref{eq:ah-mass-budget} and
\eqref{eq:ah-ts-application},
\[
 \int_\Omega(1-x)\Sigma
 =\int_S b_S+\int_\Omega P
       -8\pi\{p_0(h)-p_0(g)\}\le C.
\]
The constant is independent of \(M\).
Lemma~\ref{lem:ah-penalties} makes
\(\Sigma=L\) on \(\mathcal D\cap\{r\le R_s\}\),
and \(1-x\ge\delta\) there.  Consequently
\[
 \Vol_g(\mathcal D\cap\{r\le R_s\})\le C/L.
\]
Since \(W\le1+\sqrt{W^2-1}\), Cauchy--Schwarz and
\eqref{eq:ah-height-energy} give an inner weighted
volume bound \(C(L^{-1}+\sqrt{M/L})\).
For large \(M\), \(R_s>r_p\).  The asymptotically
hyperbolic volume element gives
\[
 \int_{\{r>R_s\}}\rho^{3/2}\,\dd V_g\le C/R_s,\qquad
 \int_{\{r>R_s\}}\rho^3\,\dd V_g\le C/R_s^4 ,
\]
where \(r_p\) is fixed and absorbed into \(C\).
A second application of Cauchy--Schwarz yields
\begin{equation}\label{eq:ah-bad-volume}
 \int_{\mathcal D}\rho^{3/2}W\,\dd V_g
 \le C\left(L^{-1}+\sqrt{M/L}
                  +R_s^{-1}+\frac{\sqrt M}{R_s^2}\right)
 \longrightarrow0 .
\end{equation}
For the choices \eqref{eq:ah-parameter-scales},
all four powers tend to zero.

We check the hypotheses of the area-transfer proposition
explicitly.  The capped smooth manifold \(X\) and cap
set \(O\) are fixed, so the relevant second Betti number
of \(X\setminus\operatorname{int}O\) is finite and fixed.
The chosen \(E\) is a cap-only largest minimizing
enclosure, has a free smooth compact minimal boundary,
contains \(O\) compactly, is strictly outward minimizing,
and each component meets \(O\).  The comparison metric
is the smooth \(h=x\bar g\); here there is no seam or
smoothing error.  It obeys
\(\bar g\ge g\), \(\dd V_{\bar g}=W\,\dd V_g\), and
\(\Scal_h\ge-6\) outside \(E\).
Its asymptotically hyperbolic end supplies the proper
subsolution required in Proposition~\ref{prop:area-transfer}.
Lemma~\ref{lem:area-projection} gives, on every relevant
enclosing level with connected exterior,
\[
 \int\rho\,\dd A_g\ge(1-o_{r_p}(1))A_*.
\]
Choose \(r_p\) first to make this error sufficiently
small for the fixed \(\varepsilon_*\), then choose
\(\delta\), and finally take \(M\) large.
The proposition's temporary conformal modification
for starting the flow may be arbitrarily small at
each fixed metric and leaves the initial area
unchanged.  Its uniform scalar lower bound therefore
has the required fixed constant, for example \(7\).
Equation~\eqref{eq:ah-bad-volume} meets the remaining
smallness hypothesis.  Proposition~\ref{prop:area-transfer}
gives
\begin{equation}\label{eq:ah-area-conclusion}
 A_h\ge(1-\varepsilon_*)A_* .
\end{equation}

Combine \eqref{eq:ah-mass-budget},
\eqref{eq:ah-ts-application}, and
\eqref{eq:ah-area-conclusion}.  For every fixed
\(\varepsilon_*>0\) and arbitrarily large \(M\),
\[
 p_0(g)+o(1)\ge F\bigl((1-\varepsilon_*)A_*\bigr).
\]
First let \(M\to\infty\), then
\(\varepsilon_*\downarrow0\), using continuity of \(F\).
If \(A_*=0\), no area transfer is needed:
\(F(A_h)\ge F(0)\) and the mass budget give the
same conclusion directly.  This proves
Theorem~\ref{thm:ah-reduction}.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/05-maximal-reduction.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/06-stress-extension.tex ====
\section{Electrostatic stress and the asymptotically flat attachment}
\label{sec:stress-extension}

Throughout this section, $(\Omega,g)$ is a smooth connected exterior with
the topology, completeness, spherical end, tensor decay, and weighted
scalar integrability specified in Section~\ref{sec:hypotheses}.  We assume
\begin{equation}
 R_g\ge-6,\qquad H_g(S)\le0,
 \label{eq:stress-base-assumptions}
\end{equation}
and make no causal assumption on its mass covector.  The normal on $S$
points into $\Omega$.  Fix the end chart, and put
\begin{equation}
 A_* = A_{\min}(S;g),\qquad a=\sqrt{A_*/(4\pi)}.
 \label{eq:stress-area-radius}
\end{equation}
The objective is to construct a compact inner region and a scalar-flat
\AF\ exterior whose mass is at most $p_0(g)-a^3/2$, up to an arbitrarily
small error.  The tensor introduced for this purpose is synthetic: it is
not the second fundamental form of the original data.  We denote the
electrostatic stress by $\Pi$, reserving $\pi$ for the graph second form in
the subsequent construction.

\subsection{A conformal approximation of the end}

\begin{proposition}[End approximation]
\label{prop:conformal-approximation}
Under \eqref{eq:stress-base-assumptions}, there are smooth metrics $g_j$
on the same exterior such that $R_{g_j}>-6$, $H_{g_j}(S)\le0$,
$g_j\to g$ locally in $C^2$, and
\begin{equation}
 (1-\epsilon_j)g\le g_j\le(1+\epsilon_j)g,
 \quad \epsilon_j\downarrow0,
 \qquad p_0(g_j)\longrightarrow p_0(g).
 \label{eq:stress-density-convergence}
\end{equation}
Each $g_j$ has the required weighted scalar integrability and finite mass
fluxes.  Outside a compact set, depending on $j$, it is of the form
\begin{equation}
 g_j=\phi_j^4 b,\qquad
 \phi_j=1+\alpha_j(\omega)r^{-3}+O(r^{-3-\xi}),
 \label{eq:stress-conformal-end}
\end{equation}
where $\xi>0$ can be fixed and the remainder has the same decay after
any fixed finite number of angular and radial-logarithmic derivatives.
In particular $A_{\min}(S;g_j)\to A_{\min}(S;g)$.
\end{proposition}

\begin{proof}
Choose $j_0$ with $3/2<j_0<\tau$, where $\tau$ is the original end
decay exponent.  Extend $r$ to a smooth positive function on the compact
part.  For large $B$, let $\chi_B$ be a smooth radial cutoff equal to one
for $r\le B$ and zero for $r\ge2B$, with a fixed profile in $r/B$.
Set
\[
 g_B=b+\chi_B(g-b)
\]
on the end and $g_B=g$ on the compact part.  For sufficiently large $B$
this is positive definite, and its uniform metric charts and
$C^{2,\alpha}$ bounds can be chosen independently of $B$.  Fix a smooth
$k>0$ equal to $r^{-4}$ far out.  We seek a small $u$ with
$\partial_\nu u=0$ on $S$ satisfying
\begin{equation}
 -8\Delta_{g_B}u+(R_{g_B}+6)(1+u)
 +6\big((1+u)^5-(1+u)\big)
 =\chi_B(R_g+6)+\delta k.
 \label{eq:stress-density-equation}
\end{equation}
The constant-term discrepancy, excluding $\delta k$, is supported in
$B<r<2B$ and is $O(r^{-\tau})$ in local $C^{0,\alpha}$ norms.
Consequently it tends to zero in the scalar weighted norm of weight
$r^{-j_0}$.  Also $R_{g_B}+6\ge-o(1)$ uniformly, because it equals the
nonnegative $R_g+6$ on the inner part, vanishes beyond $2B$, and is
$O(B^{-\tau})$ on the intervening annulus.

For completeness, the linearized operator divided by eight is
\[
 \mathscr L_B=-\Delta_{g_B}+3+(R_{g_B}+6)/8.
\]
It has an inverse, with homogeneous Neumann condition, bounded uniformly
from weighted $C^{0,\alpha}$ to weighted $C^{2,\alpha}$ at weight
$j_0$ (a smaller H\"older exponent may be used).  To prove the required
supremum bound, take a positive function $\beta$ constant on a large
compact set, comparable to $r^{-j_0}$ at infinity, and let its logarithmic
slope decrease slowly from zero to $-j_0$ as a function of $\log r$.
For the limiting radial operator,
\[
 (-\partial_\eta^2-2\partial_\eta+3)e^{-j_0\eta}
 =(3+2j_0-j_0^2)e^{-j_0\eta}>0.
\]
Every intermediate slope has the same strict sign.  Slow variation of
the slope, placement of the transition sufficiently far out, and the
uniformly small negative part of $(R_{g_B}+6)/8$ therefore give
$\mathscr L_B\beta\ge c\beta$ with $c>0$ independent of large $B$.
Here $\partial_\nu\beta=0$ on $S$.  Solve on compact truncations with
zero outer Dirichlet data.  The maximum principle, applied to positive
and negative multiples of $\beta$, bounds the solution by the weighted
norm of its forcing.  Uniform local interior and Neumann Schauder
estimates give the stated second-order bound.  Compactness gives a
solution on $\Omega$, and the maximum principle gives uniqueness.
This proves the inverse assertion without an assumption about solvability
on the noncompact manifold.

The nonlinear remainder in \eqref{eq:stress-density-equation} is
$O(u^2)$ and is locally Lipschitz with Lipschitz constant tending to zero
on small weighted balls.  The uniform inverse and contraction mapping
therefore give $u=u_{B,\delta}$ tending to zero in weighted
$C^{2,\alpha}_{j_0}$ as $B\to\infty$ and $\delta\downarrow0$.
Local elliptic bootstrapping makes $u$ smooth.  For small parameters
$1+u>0$, and the conformal scalar-curvature formula gives, for
$\widetilde g=(1+u)^4g_B$,
\begin{equation}
 R_{\widetilde g}+6
 =(1+u)^{-5}\big(\chi_B(R_g+6)+\delta k\big)>0.
 \label{eq:stress-density-scalar}
\end{equation}
The boundary conformal formula and $\partial_\nu u=0$ give
$H_{\widetilde g}=(1+u)^{-2}H_g\le0$.  The uniform metric convergence in
\eqref{eq:stress-density-convergence} follows from the weighted bound and
the cutoff construction.

We next justify the end expansion for each fixed $B,\delta$.
Beyond $2B$, $g_B=b$ and
\begin{equation}
 (-\Delta_b+3)u=F(u)+\delta r^{-4}/8,
 \qquad F(u)=O(u^2),\quad F'(u)=O(u).
 \label{eq:stress-density-end-ode}
\end{equation}
All fixed orders of angular derivatives of $u$ are $O(r^{-j_0})$.
Indeed rotations commute with $\Delta_b$.  Their difference quotients
solve a linear equation with small potential $O(u)$, with smooth data
on a fixed inner sphere and decaying data at infinity.  The same
$r^{-j_0}$ barriers and local weighted estimates bound these quotients.
Induction, with the lower angular derivatives as forcing, proves the
claim at every fixed order.  Radial derivatives at the initial weighted
order follow from local elliptic estimates.  Since
\[
 \Delta_b=\partial_\eta^2+2\coth\eta\,\partial_\eta
            +r^{-2}\Delta_\sigma,
\]
Equation~\eqref{eq:stress-density-end-ode} reduces, with all these angular
derivatives, to
\[
 -u_{\eta\eta}-2u_\eta+3u=O(e^{-(3+\xi)\eta})
\]
for any fixed sufficiently small
$0<\xi<\min\{2j_0-3,j_0-1,1\}$.  Variation of constants for the two
solutions $e^\eta,e^{-3\eta}$ excludes the growing mode by the known
decay and gives $u=c(\omega)e^{-3\eta}+O(e^{-(3+\xi)\eta})$.
The same argument after angular differentiation makes $c$ smooth;
the equation gives all stated radial derivatives.  Changing from
$\eta$ to $r$ gives \eqref{eq:stress-conformal-end}.

We finally check mass convergence, since convergence only in a weight
less than three would not suffice.  Write $h=b+e$ for either the original
metric or one of the approximants, and let $\mathbb U_{V_0}(e)$ be the
one-form in the mass definition.  The linearized scalar curvature is
\[
 (DR)_b(e)=\divg_b\divg_b e-\Delta_b\tr_b e+2\tr_b e.
\]
Differentiation, using $\nabla_b^2V_0=V_0b$, yields the exact linear
identity $\divg_b\mathbb U_{V_0}(e)=V_0(DR)_b(e)$.
The difference between $R_h+6$ and $(DR)_b(e)$ is bounded by
$C(|e|\,|\nabla_b^2e|+|\nabla_be|^2+|e|^2)$.  The uniform weighted
second-order bound and $dV_b\asymp r\,dr\,dA_\sigma$ thus give
\begin{equation}
 16\pi p_0(h)-\int_{r=D}\mathbb U_{V_0}(h-b)(\nu_b)\,\dd A_b
 =\int_{r>D}V_0(R_h+6)\,\dd V_b+O(D^{3-2j_0}),
 \label{eq:stress-density-mass-tail}
\end{equation}
uniformly in the approximants.  The scalar integrands in
\eqref{eq:stress-density-scalar} are dominated by a fixed multiple of
$V_0(R_g+6+k)$, which is integrable; they converge pointwise to
$V_0(R_g+6)$.  The same divergence calculation with $V_i$, using
$|V_i|\le V_0$ and the corresponding derivative bounds, proves existence
of all fluxes for the approximants.  At a fixed $D$ their surface fluxes
converge by local $C^2$ convergence.  Dominated convergence in
\eqref{eq:stress-density-mass-tail}, followed by $D\to\infty$, proves
the asserted time-component convergence.  This is also the scalar
linearization underlying the mass definition in \citet{CH2003}.
Finally, the two-sided metric comparison compares the areas of every
admissible intrinsic boundary with the same multiplicative factors;
taking infima proves convergence of the minimum enclosing areas.
\end{proof}

We henceforth fix one approximating metric, write it again as $g$, and
write $\alpha$ for its smooth coefficient in
\eqref{eq:stress-conformal-end}.  Thus $R_g+6>0$ everywhere.  All
subsequent constants may depend on this fixed metric; no uniformity
across the approximation is needed.

\subsection{The nonlinear electrostatic problem}

For $0\le e\le1$ define
\begin{equation}
 t(e)=\sqrt{1-e^{3/2}},\qquad
 L(w)=\max_{0\le e\le1}\{t(e)+we\},\qquad w\ge0.
 \label{eq:stress-legendre-law}
\end{equation}
Strict concavity of $t$, together with $t'(0)=0$ and
$t'(e)\to-\infty$ as $e\uparrow1$, gives a unique maximizing $e=e(w)$,
with $e(0)=0$, $0<e(w)<1$ for $w>0$, and
\begin{equation}
 w=-t'(e)=\frac{3\sqrt e}{4t(e)},\qquad
 L'(w)=e(w),\qquad L=t+we.
 \label{eq:stress-constitutive-law}
\end{equation}
In particular $L(|\xi|)$ is convex as a function of a covector $\xi$.

\begin{proposition}[Electrostatic potential, stress, and charge]
\label{prop:electrostatic-stress}
For every $M_e>0$ there is a weak solution $u$ of
\begin{equation}
 \divg_g E=0,\qquad
 E=e(w)\frac{\nabla u}{w},\quad w=|du|_g,
 \qquad u|_S=-M_e,\quad u\longrightarrow0\text{ at infinity},
 \label{eq:stress-electrostatics}
\end{equation}
where $E=0$ at $w=0$.  It satisfies $-M_e\le u\le0$,
$|u|+|du|_g=O(r^{-1})$, and is $C^{1,\beta}$ locally up to $S$ and
in compactified coordinates $(\rho,\omega)$ up to $\rho=0$, for some
$\beta>0$, where $\rho=e^{-\eta}$.
The continuous symmetric tensor
\begin{equation}
 \Pi=L(w)g-du\otimes E^\flat
 \label{eq:stress-tensor}
\end{equation}
is distributionally divergence free, is positive definite, and obeys
\begin{equation}
 (\tr_g\Pi)^2-2|\Pi|_g^2=3.
 \label{eq:stress-algebraic-identity}
\end{equation}
There is a continuous nonnegative function $e_0$ on $\Sph^2$ such that,
uniformly in angle,
\begin{equation}
 r^2|E|_g\longrightarrow e_0,\qquad
 r^3\big(\Pi(\nu,\nu)-1\big)
       \longrightarrow-\tfrac12 e_0^{3/2}.
 \label{eq:stress-end-asymptotics}
\end{equation}
Here $\nu$ points toward increasing $r$.  The charge
\begin{equation}
 F_{M_e}=\int_{r=R}g(E,\nu)\,\dd A_g
 \label{eq:stress-charge-definition}
\end{equation}
is independent of large $R$, satisfies
$F_{M_e}\le\int_{\Sph^2}e_0\,\dd A_\sigma$, and
\begin{equation}
 \liminf_{M_e\to\infty}F_{M_e}\ge A_*.
 \label{eq:stress-charge-saturation}
\end{equation}
\end{proposition}

\begin{proof}
We give the construction and its regularity before establishing the
stress and charge assertions.

\emph{Constitutive estimates and regularization.}
For $w>0$ put $Q(w)=we'(w)/e(w)$.  Logarithmic differentiation of
\eqref{eq:stress-constitutive-law} gives
\begin{equation}
 Q(w)^{-1}=\frac12+\frac{3e(w)^{3/2}}{4(1-e(w)^{3/2})}.
 \label{eq:stress-Q-formula}
\end{equation}
Thus $0<Q\le2$.  At zero, inversion in the variable $\sqrt e$ gives
\[
 e(w)=\frac{16}{9}w^2+O(w^5);
\]
in particular $e(w)/w^2$ extends smoothly and positively to $w=0$.
At infinity, inversion in $w^{-2}$ gives
\begin{equation}
 e(w)=1-\frac{3}{8w^2}+O(w^{-4}),\qquad
 Q(w)=\frac{3}{4w^2}+O(w^{-4}),\qquad
 wQ'(w)=-\frac{3}{2w^2}+O(w^{-4}).
 \label{eq:stress-large-gradient-law}
\end{equation}
Regularize only at zero by
\begin{equation}
 e_\varepsilon(w)=w\frac{e(z)}z,\quad
 z=(w^2+\varepsilon^2)^{1/2},\quad
 Q_\varepsilon(w)=1+\frac{w^2}{z^2}(Q(z)-1),\quad 0<\varepsilon<1.
 \label{eq:stress-regularized-law}
\end{equation}
The flux $e_\varepsilon(|\xi|)\xi^\sharp/|\xi|$ is smooth and strictly
elliptic on every finite gradient range, including zero.  Its radial
and transverse derivative eigenvalues are $e_\varepsilon'$ and
$e_\varepsilon/w$, respectively.  We have $0<Q_\varepsilon\le2$;
uniformly for sufficiently large $w$,
\begin{equation}
 c\le w^2Q_\varepsilon(w)\le C,\qquad
 -w^2\big(wQ_\varepsilon'(w)+Q_\varepsilon(w)\big)\ge c.
 \label{eq:stress-uniform-Q}
\end{equation}
These follow by substituting \eqref{eq:stress-large-gradient-law} in
\eqref{eq:stress-regularized-law}; the leading coefficient is
$3/4+\varepsilon^2$, bounded above and below independently of
$\varepsilon$.

\emph{Boundary and interior gradient bounds.}
On a connected smooth truncation $\Omega_R$ prescribe $u=-M_e$ on $S$
and $u=0$ on $r=R$.  The maximum principle gives $-M_e\le u\le0$.
At points with nonzero gradient the regularized equation is
\begin{equation}
 P:\nabla^2u=0,\qquad
 P=g^{-1}+(Q_\varepsilon(w)-1)n\otimes n,
 \quad n=\nabla u/w.
 \label{eq:stress-regularized-nondivergence}
\end{equation}
Let $h$ be inward distance from $S$.  Since $H(S)\le0$, smoothness
gives $\Delta h\le C h$ in a fixed collar.  A supersolution above the
inner boundary datum is $-M_e+B(h)$, where
\[
 B(h)=b_0^{-1}\log(1+Dh),\qquad B''=-b_0(B')^2.
\]
For $p=B'$ sufficiently large, \eqref{eq:stress-uniform-Q} gives
\[
 P:\nabla^2B=Q_\varepsilon(p)B''+p\Delta h
 \le-cb_0+C/b_0,
\]
because $hp\le1/b_0$.  Choose $b_0$ first to make this negative,
then a collar width $h_0$ so small that
$(2b_0h_0)^{-1}$ is above the required gradient threshold, and finally
$D$ large so that $B(h_0)>M_e$ and $B'\ge(2b_0h_0)^{-1}$ there.
Comparison with the constant lower barrier bounds the inward derivative
on $S$, independently of $R$ and $\varepsilon$.
At the outer sphere, inward distance may be replaced by
$h=\eta_R-\eta$ in a uniform collar.  For gradients parallel to
$d\eta$, asymptotic hyperbolicity and $Q_\varepsilon\le2$ give
$P:\nabla^2h\le-1$ for sufficiently large $R$.  The linear lower
barrier $-C h$, together with $u\le0$, bounds the outer derivative
with $C$ depending on $M_e$ and the fixed collar width only.

For the interior estimate use an orthonormal frame with first vector
$n$, and write transverse indices as $a,b$.  Differentiating
\eqref{eq:stress-regularized-nondivergence}, commuting one derivative,
and differentiating the norm of $du$ gives
\begin{align}
 P:\nabla^2\log w
 ={}&\Ric(n,n)
 +(1-Q_\varepsilon)\frac{|d_\perp w|^2}{w^2}
 +\frac{\sum_{a,b}u_{ab}^2}{w^2} \notag\\
 &-\big(wQ_\varepsilon'+Q_\varepsilon\big)\frac{w_1^2}{w^2}.
 \label{eq:stress-gradient-identity}
\end{align}
To specify the cancellation, the differentiated coefficient term is
$Q_\varepsilon'w_1^2+2(Q_\varepsilon-1)|d_\perp w|^2/w$;
the differentiated norm contributes
$[Q_\varepsilon\sum_a u_{1a}^2+\sum_{a,b}u_{ab}^2]/w$ before
division by $w$.  Subtracting $P(dw,dw)/w^2$ produces exactly
\eqref{eq:stress-gradient-identity}.
At a positive interior maximum of $w\exp(B_0u)$ we have
$d_\perp w=0$ and $w_1/w=-B_0w$.  The left side of
\eqref{eq:stress-gradient-identity} is nonpositive, since
$P:\nabla^2u=0$.  The global lower Ricci bound and
\eqref{eq:stress-uniform-Q} make the right side at least
$-C+cB_0^2$ when $w$ is large.  A fixed sufficiently large $B_0$
excludes such a maximum.  Combining the remaining interior maxima
with the boundary estimates and the height bound gives
\begin{equation}
 \sup_{\Omega_R}|du|_g\le C(M_e),
 \label{eq:stress-global-gradient-bound}
\end{equation}
uniformly in large $R$ and $0<\varepsilon<1$.

\emph{Existence on truncations and passage to the limit.}
For fixed $R,\varepsilon$, vary the inner Dirichlet value from zero
to $-M_e$.  The derivative of the divergence equation is a strictly
elliptic divergence operator without zeroth-order term and is
invertible with zero Dirichlet data by the maximum principle and
linear elliptic theory.  The bounds already proved hold on this
entire path.  On its bounded gradient range, ordinary uniformly
elliptic quasilinear gradient H\"older estimates apply.  For the
boundary estimate subtract its constant value and reflect oddly in
boundary normal coordinates, reflecting the metric as well.  The
reflected metric is Lipschitz; its mixed normal--tangential entries
vanish on the face.  The reflected equation is weakly valid by the
zero trace.  Equivalently, difference quotients give uniformly
elliptic equations for the first derivatives with bounded divergence
forcing.  Scalar H\"older estimates followed by Schauder estimates
on the original smooth side give closedness of the continuation.
Here constants may depend on the fixed $R,\varepsilon$;
\citet{GT2001} supplies the ordinary elliptic estimates being used.
Thus smooth regularized solutions exist.

A decay bound is uniform in both regularizations.  On the end put
$v=C(e^{-\eta}-e^{-2\eta})$.  It is positive for large $\eta$, and
\[
 \frac{v''}{-v'}=\frac{1-4e^{-\eta}}{1-2e^{-\eta}}
 \le1-e^{-\eta}.
\]
Also $|d\eta|_g^2=1+O(e^{-\tau\eta})$ and
$P:\nabla^2\eta=2+O(e^{-\min\{\tau,2\}\eta})$ for the
radial test direction, uniformly for $0<Q_\varepsilon\le2$.
Since $\tau>1$, these inequalities imply
$P:\nabla^2v\le0$ sufficiently far out, independently of $C$.
Comparison of the nonnegative solution $-u$ with $v$, choosing $C$
on a fixed inner sphere and using $u=0$ on the truncating edge,
gives $|u|\le Ce^{-\eta}$.

Take a subsequence with $R\to\infty$ and $\varepsilon\downarrow0$.
The uniform Lipschitz bound gives local uniform convergence and weak
gradient convergence to a locally Lipschitz function with the required
traces.  To pass to the equation, let $\chi$ be a compactly supported
cutoff and test the regularized equation with $\chi(u_j-u)$.
The terms containing $d\chi$ tend to zero by uniform convergence.
Subtracting the limiting-law flux evaluated at $du$ gives
\[
 \int\chi\,
 \big(A(du_j)-A(du)\big)\cdot(du_j-du)\,\dd V_g\longrightarrow0,
 \qquad A(\xi)=e(|\xi|)\xi^\sharp/|\xi|.
\]
Uniform convergence of the constitutive functions on bounded ranges
justifies replacing the regularized flux here.  The law $A$ is
strictly monotone.  On the fixed compact gradient range its
monotonicity pairing has a positive minimum whenever the two
arguments are separated by a prescribed positive distance.
Thus $du_j\to du$ in measure and strongly in every finite local
$L^p$.  This proves the limiting weak equation.

\emph{Gradient regularity and compactification.}
We apply the scalar $p=3$ gradient theorem recorded in
Section~\ref{sec:preliminaries}, namely Theorem~1.1 and estimate~(1.8)
of \citet{AraujoZhang2020}.  The coordinate flux must satisfy globally
\begin{align}
 \lambda|\xi|\,|\zeta|^2
 &\le \mathcal A_\xi(x,\xi)[\zeta,\zeta],
 &|\mathcal A|+|\mathcal A_\xi|\,|\xi|&\le C|\xi|^2,
 \label{eq:stress-p3-structure}\\
 |\mathcal A(x,\xi)-\mathcal A(y,\xi)|
 &\le C|x-y|\,|\xi|^2.
 \notag
\end{align}
It must also be $C^1$ in $\xi$ and the solution must have bounded
local $W^{1,3}$ energy.  The Lipschitz spatial bound supplies a
$C^{0,1/2}$ modulus, so $n=p=3$ and zero forcing meet all the
parameters of that theorem.  Throughout this argument choose only
$0<\beta<\min\{\alpha_m,1/4\}$ as in
Section~\ref{sec:preliminaries}; the decay and boundary limits below
require a positive exponent, not an optimal one.

We verify global, rather than merely bounded-gradient, structure before
applying this theorem.  On the gradient range in
\eqref{eq:stress-global-gradient-bound}, $e(w)/w^2$ and $Q(w)$ have
positive upper and lower bounds.  Thus $e'(w)$ is comparable to $w$.
Choose a smooth cutoff $\chi=1$ below the known range and zero above
twice a larger fixed bound, and replace $e'$ outside that range by
\[
 \widetilde e'(w)=\chi(w)e'(w)+(1-\chi(w))c_1w,
 \qquad \widetilde e(0)=0,
\]
where $c_1>0$.  Then $\widetilde e=e$ on the range attained by the
solution, $\widetilde e'\asymp w$, and $\widetilde e\asymp w^2$
globally.  In coordinates with metric matrix $G$ the density flux is
\begin{equation}
 \mathcal A^i(x,\xi)=\sqrt{\det G}\,
 \frac{\widetilde e(|\xi|_G)}{|\xi|_G}\,G^{ij}\xi_j,
 \qquad |\xi|_G=(G^{ij}\xi_i\xi_j)^{1/2}.
 \label{eq:stress-coordinate-flux}
\end{equation}
Uniform positive metric bounds give
\eqref{eq:stress-p3-structure}; Lipschitz metric coefficients give
the spatial bound.  The flux is $C^1$ at zero because it is
$O(|\xi|^2)$ and its derivative is $O(|\xi|)$.  Its derivative is
symmetric, as is also apparent from its convex radial primitive.
The solution has finite local cubic energy by its Lipschitz bound.
The quoted theorem therefore proves local $C^{1,\beta}$ regularity.

The same argument works at $S$.  In orthogonal boundary coordinates
with reflection matrix $J=\operatorname{diag}(-1,1,1)$, extend the
metric matrix by $G(-h,x)=JG(h,x)J$, and extend $u+M_e$ oddly.
The mixed metric entries vanish on the face, so this extension is
Lipschitz.  The reflected flux satisfies
$\mathcal A_-(\xi)=J\mathcal A_+(J\xi)$.  Evaluated on the odd
solution, its normal component is even and its tangential components
are odd.  A test on the reflected ball consequently reduces to a
test on the original half-ball with zero trace.  Such tests are
allowed: cutoff at distance $h=\delta$ creates an error tending to
zero since the test is $O(h)$, the flux is bounded, and the strip
volume tends to zero.  Thus the reflected weak equation satisfies
all the same hypotheses, proving the asserted boundary regularity.

To obtain gradient decay, take a uniform metric ball far out with
central radius $r_*$, and put $s_*=C/r_*$, so that $|u|\le s_*$ on a
slightly larger ball.  Apply the preceding fixed global extension
to the equation for $u/s_*$; its flux is
\[
 \mathcal A_{s_*}(x,\xi)=s_*^{-2}\mathcal A(x,s_*\xi).
\]
The cubic structure constants and spatial modulus are unchanged.
The required energy bound does not presume a bound on the normalized
gradient.  Testing with $\chi^3(u/s_*)$, and using coercivity and
Young's inequality, gives
\begin{equation}
 \int\chi^3|d(u/s_*)|^3
 \le C\int |u/s_*|^3|d\chi|^3\le C.
 \label{eq:stress-scaled-energy}
\end{equation}
The H\"older estimate for the gradient, together with this energy
bound, gives its supremum on a smaller ball: some point has gradient
bounded by the cubic mean, and the H\"older bound controls its
difference from every other point.  Therefore $|du|_g\le C/r$.

Now let $g_c=\rho^2g$ on the end.  It extends to a positive Lipschitz
metric up to $\rho=0$, with vanishing mixed normal--tangential entries
there.  In the conformal end at hand this follows directly from
\[
 g_c=\phi^4\left(d\rho^2+\tfrac14(1-\rho^2)^2\sigma\right)
\]
and \eqref{eq:stress-conformal-end}.  The estimate just obtained gives
$|du|_{g_c}\le C$, and the height estimate gives the zero trace.
Writing $\bar w=|du|_{g_c}$, the coordinate divergence equation in
this metric has radial law
\begin{equation}
 q_\rho(\bar w)=\rho^{-2}e(\rho\bar w)
              =\bar w^2 F(\rho\bar w),
 \qquad F(t)=e(t)/t^2.
 \label{eq:stress-compactified-law}
\end{equation}
On the known bounded range of $\bar w$, this satisfies uniform cubic
structure and Lipschitz spatial dependence, including at $\rho=0$.
For an explicit uniform extension, set
$H(t)=2F(t)+tF'(t)>0$, choose a fixed cutoff equal to one below that
range and zero above twice a larger bound, and define
\[
 \widetilde q_\rho'(w)
 =w\big[\chi(w)H(\rho w)+(1-\chi(w))c_1\big],
 \qquad \widetilde q_\rho(0)=0.
\]
It agrees with $q_\rho$ on the attained range, is comparable to
$w^2$, has derivative comparable to $w$, and has the required uniform
Lipschitz dependence in position.  Reflect $g_c$ as above, use
$|\rho|$ in this coefficient, and reflect $u$ oddly.  The gradient
bound ensures $W^{1,3}$ across the face.  The same zero-trace test
argument proves the reflected weak equation.  The scalar theorem
now gives $C^{1,\beta}$ in compactified coordinates.  No theorem
about degenerate boundary equations is being assumed in addition
to the stated interior result.

\emph{Stress and asymptotic coefficients.}
Convexity shows that the weak solution minimizes the integral of
$L(|du|)$ against compactly supported Lipschitz variations: integrate
the convex tangent inequality and use \eqref{eq:stress-electrostatics}.
Apply this fact to compactly supported changes of independent variable.
For the flow of a smooth vector field $X$, differentiating the local
energy of the transported potential gives
\[
 \int\big(L(w)g-du\otimes E^\flat\big):\nabla X\,\dd V_g=0.
\]
The calculation is justified by bounded gradients on the support;
no finite total energy of $L$ is needed.  This is
$\divg_g\Pi=0$ in distributions.
The stress eigenvalues in the gradient and transverse directions are
$t$ and $l=L=t+we$, respectively, with $\Pi=g$ at zero gradient.
They are positive at every finite gradient.  Moreover,
\[
 (t+2l)^2-2(t^2+2l^2)=4tl-t^2
 =3t^2+4twe=3(1-e^{3/2})+3e^{3/2}=3,
\]
proving \eqref{eq:stress-algebraic-identity}.

Since $u$ has constant trace at conformal infinity, all its tangential
derivatives in compactified coordinates vanish there.  Its scaled
gradient consequently has a continuous radial limit.  Together with
$r\rho\to1/2$ and $e(w)=(16/9)w^2+O(w^5)$, this gives a continuous
$e_0\ge0$ with $r^2e(w)\to e_0$ uniformly.  At a point where
$e_0>0$ the gradient direction approaches the radial direction, so
$\Pi(\nu,\nu)=t+o(r^{-3})$.  At a point where $e_0=0$, both
$t-1$ and $l-t=we$ are $O(e^{3/2})=o(r^{-3})$, and the same conclusion
holds.  Uniformity follows by splitting into the sets where $e_0$
is larger or smaller than an arbitrary fixed positive threshold.
Since $t=1-e^{3/2}/2+O(e^3)$, this proves
\eqref{eq:stress-end-asymptotics}, as well as
$E=O(r^{-2})$ and $\Pi-g=O(r^{-3})$ in metric norm.

\emph{Flux saturation.}
Continuous distributionally divergence-free fields have the usual
flux identities on smooth compact domains: use smooth cutoffs
approaching their boundaries and continuity in the shrinking collars.
This proves independence in \eqref{eq:stress-charge-definition}.
The asymptotics and $dA_g/r^2\to dA_\sigma$ give
$F_{M_e}\le\int e_0\,dA_\sigma$.  Integration by parts in the
electrostatic equation gives
\begin{equation}
 M_eF_{M_e}=\int_\Omega w e(w)\,\dd V_g.
 \label{eq:stress-energy-charge}
\end{equation}
Indeed the infinity term tends to zero because
$u=O(r^{-1})$, $E=O(r^{-2})$, and the sphere area is $O(r^2)$.
The inner term has the indicated sign: the outward normal of the
exterior at $S$ is $-\nu$, while $u=-M_e$ there.  The integral is
finite since $we=O(r^{-3})$ and $dV_g\asymp r\,dr\,dA_\sigma$.

Fix a large $R_0$, choose $4/3<s_0<2$, and set
\begin{equation}
 \zeta=1\quad(r\le R_0),\qquad
 \zeta=(R_0/r)^{s_0}\quad(r>R_0),\qquad
 C_\zeta=\int_\Omega(1-t(\zeta))\,\dd V_g<\infty.
 \label{eq:stress-charge-weight}
\end{equation}
The last integrability follows from
$1-t(\zeta)=O(\zeta^{3/2})$ and $s_0>4/3$.
For every smooth function $v$ equal to $-M_e$ near $S$ and zero
outside a compact set, each regular level $v=q$, $-M_e<q<0$, obeys
\begin{equation}
 \int_{v=q}\zeta\,\dd A_g
 \ge(1-o_{R_0}(1))A_*.
 \label{eq:stress-weighted-level-area}
\end{equation}
Here is the topological and geometric comparison for this assertion.
Take the closure $D$ of the component of $\{v>q\}$ containing the
end.  It is a connected closed end-side domain, with boundary a
subset of the level; it misses a neighborhood of all of $S$.
Choose a sphere $r=R_1$, with $R_1\in[R_0/2,R_0]$, transverse to its
boundary, and adjoin to $D$ the whole region $r\ge R_1$.
The union is still connected because the two domains have a common
far-end region.  Each newly exposed sphere-cap point lies on a
radial ray meeting the old boundary at some $r\ge R_1$: initially
the ray is outside $D$, and eventually it lies in $D$.
Radial projection onto $r=R_1$ has upper two-dimensional Jacobian
\[
 (1+o_{R_0}(1))(R_1/r)^2
 \le(1+o_{R_0}(1))\zeta(r).
\]
The first estimate is uniform over tangent two-planes by the
asymptotic tensor bounds; the second uses $s_0<2$ and $R_1\le R_0$.
The area formula bounds the new caps by the weighted old boundary
outside the sphere.  Inside, the weight is one.  The transverse
corners can be rounded toward the side enlarging this union, in
shrinking tubular neighborhoods of their intersection curves; the
added area tends to zero.  The rounded exterior is connected and
its full intrinsic boundary is an admissible cut in the definition
of $A_*$.  This proves \eqref{eq:stress-weighted-level-area} without
any connectedness assertion about the level itself.

Coarea therefore gives
$\int\zeta|dv|\ge M_e(1-o_{R_0}(1))A_*$.
The same holds for $u$.  To justify approximation, first replace
$u$ by its constant trace in a shrinking collar of $S$, interpolating
in a collar of comparable width.  The error in its weighted gradient
integral tends to zero by the bounded gradient and constant trace.
Next multiply by a cutoff tending to zero in $R<r<2R$ and set it
equal to zero beyond $2R$.  The height and gradient decay make both
the lost tail and cutoff error $O(R^{1-s_0})$, tending to zero.
Smooth the remaining compactly supported transition while retaining
the constant boundary neighborhoods.  This gives convergence in
the required weighted $W^{1,1}$ norm, and coarea applies to the
smooth approximants.

Finally, the definition of $L$ gives $L(w)\ge\zeta w+t(\zeta)$,
whereas $L(w)-1\le we(w)$ since $t(e)\le1$.  Combining these facts
with \eqref{eq:stress-energy-charge} yields
\begin{equation}
 F_{M_e}\ge(1-o_{R_0}(1))A_*-C_\zeta/M_e.
 \label{eq:stress-charge-lower-bound}
\end{equation}
Let $M_e\to\infty$ at fixed $R_0$ and then $R_0\to\infty$.
This proves \eqref{eq:stress-charge-saturation} and completes the
proposition.
\end{proof}

\subsection{Smoothing the synthetic tensor and attaching an exterior}

\begin{proposition}[Compact stress data and AF attachment]
\label{prop:af-attachment}
Let $g$ be the fixed approximating metric above.  For every
$\varepsilon_0>0$ there are a sufficiently large coordinate sphere
$\mathcal C$, its compact connected inner region $\Omega_i$, a
smooth symmetric tensor $K$ on $\Omega_i$, and a smooth scalar-flat
\AF\ exterior $(\Omega_o,g_o)$ attached along $\mathcal C$, with
the following properties.  Put $\tau_K=\tr_g K$ and
$k_T=\tr_{\mathcal C}K$.  Then
\begin{align}
 K&<0,\qquad
 R_g+\tau_K^2-|K|_g^2
   -2(\divg_gK-d\tau_K)\cdot v>0
       &&(|v|_g\le1),
 \label{eq:stress-synthetic-DEC}\\
 H_i&>|k_T|,\qquad
 H_o=\sqrt{H_i^2-k_T^2}>0,
 \label{eq:stress-matching-curvatures}\\
 m_o&\le p_0(g)-\tfrac12 a^3+\varepsilon_0.
 \label{eq:stress-attachment-budget}
\end{align}
Here $H_i$ is the inner metric's mean curvature on $\mathcal C$,
and $H_o$ is the outer metric's inner-boundary mean curvature, both
in the direction toward the respective end.  The induced metrics
agree.  The exterior is complete with its boundary included and
has one end, with $g_o-\delta=O_k(|x'|^{-1})$ for every fixed $k$,
where an ordinary derivative improves decay by one power.
Projection from $\Omega_o$ to $\mathcal C$ along its parallel rays
has upper area Jacobian at most one on every tangent two-plane.
\end{proposition}

\begin{proof}
Choose an electrostatic height $M_e$ large enough that the preceding
proposition gives
\begin{equation}
 \langle e_0\rangle\ge a^2-o(1),\qquad
 \langle f\rangle=(4\pi)^{-1}\int_{\Sph^2}f\,\dd A_\sigma,
 \label{eq:stress-charge-average}
\end{equation}
where the error can be arbitrarily small.  On the whole exterior
first define a continuous tensor $K^{(0)}$ by
\begin{equation}
 K^{(0)}-(\tr_gK^{(0)})g=2\Pi,
 \qquad K^{(0)}=2\Pi-(\tr_g\Pi)g.
 \label{eq:stress-synthetic-tensor}
\end{equation}
Its eigenvalues are $t-2l=-t-2we<0$ in the gradient direction and
$-t<0$ transversely.  Directly from
\eqref{eq:stress-algebraic-identity},
\begin{equation}
 (\tr_gK^{(0)})^2-|K^{(0)}|_g^2=6,
 \qquad
 \divg_g\big(K^{(0)}-(\tr_gK^{(0)})g\big)=0
 \label{eq:stress-synthetic-identities}
\end{equation}
in distributions.  For example, if $s_\Pi=\tr_g\Pi$, then
$\tr_gK^{(0)}=-s_\Pi$ and
$|K^{(0)}|^2=4|\Pi|^2-s_\Pi^2$, giving the first identity.

On a coordinate sphere of radius $r$, the induced metric is
$\phi^4r^2\sigma$, hence its Gauss curvature is positive for large
$r$.  The conformal mean-curvature formula gives
\begin{align}
 H_i
 &=\phi^{-2}\left(\frac{2\sqrt{1+r^2}}r
       +4\sqrt{1+r^2}\,\partial_r\log\phi\right)\notag\\
 &=2+r^{-2}-16\alpha r^{-3}+o(r^{-3}).
 \label{eq:stress-inner-H-expansion}
\end{align}
The relation \eqref{eq:stress-synthetic-tensor} gives exactly
$\tr_{\mathcal C}K^{(0)}=-2\Pi(\nu,\nu)$, so
\eqref{eq:stress-end-asymptotics} yields
\begin{equation}
 k_T^{(0)}=-2+e_0^{3/2}r^{-3}+o(r^{-3}).
 \label{eq:stress-tangential-K-expansion}
\end{equation}
All these remainders are uniform in angle.  Consequently
$H_i>|k_T^{(0)}|$ on every sufficiently large sphere, and
\begin{equation}
 \sqrt{H_i^2-(k_T^{(0)})^2}
 =\frac2r+\frac{-16\alpha+e_0^{3/2}}{r^2}+o(r^{-2}).
 \label{eq:stress-outer-H-expansion}
\end{equation}
Indeed the expression under the root is
$4r^{-2}+(-64\alpha+4e_0^{3/2})r^{-3}+o(r^{-3})$.

We explain carefully the smoothing used after fixing such a sphere
$\mathcal C$.  On the compact $\Omega_i$, continuity and strict
positivity give positive lower bounds for the eigenvalues of $\Pi$,
the absolute negative eigenvalues of $K^{(0)}$, and $R_g+6$.
There are smooth tensors $\Pi_h$ on $\Omega_i$, up to its boundary,
such that
\begin{equation}
 \|\Pi_h-\Pi\|_{C^0}\longrightarrow0,
 \qquad \|\divg_g\Pi_h\|_{C^0}\longrightarrow0.
 \label{eq:stress-divergence-smoothing}
\end{equation}
Here is the needed commutator proof, including the boundary.
In an interior chart convolve the tensor components with a smooth
kernel $\eta_h$.  A divergence term has form
$a(x)\partial_jT+b(x)T$, with smooth coefficients.  For the principal
part, integration by parts in the commutator produces
$(a(y)-a(x))\partial\eta_h(x-y)$ and a term with
$(\partial a)(y)\eta_h(x-y)$.  Together these cancel on constant
$T$.  After subtracting $T(x)$ their integral is bounded by a fixed
constant times the modulus of continuity of $T$ on a ball of radius
$O(h)$, and hence tends uniformly to zero.  The lower-order terms
commute up to the same type of error.

In a boundary chart evaluate the convolution instead at
$x+hc_0n_0$, with $n_0$ a constant inward coordinate direction and
$c_0$ large enough that the kernel stays on the original side.
The same proof applies: $|a(y)-a(x)|=O(h)$ throughout its support,
and the combined commutator still vanishes on a constant tensor.
It therefore extends the uniform estimate to the boundary without
extending the original stress as a divergence-free tensor.  Take
these local approximations on slightly larger charts and recombine
with a smooth partition of unity.  The extra differentiated
partition terms tend to zero, because their limiting sum is
$(\sum d\chi)\Pi=0$.  This proves
\eqref{eq:stress-divergence-smoothing}.

Set $K_h=2\Pi_h-(\tr_g\Pi_h)g$.  Then
$\divg K_h-d\tr K_h=2\divg\Pi_h\to0$ uniformly, and the algebraic
part $R_g+(\tr K_h)^2-|K_h|^2$ tends uniformly to $R_g+6>0$.
For sufficiently small $h$, \eqref{eq:stress-synthetic-DEC} follows
uniformly for $|v|\le1$, as does $K_h<0$.  The strict gap
$H_i>|k_T^{(0)}|$ at the now fixed sphere also persists.
Write $K$ for this smooth tensor.  The error in any integral
involving $\sqrt{H_i^2-k_T^2}$ can be made arbitrarily small at this
fixed sphere, because the square root is smooth on its positive
gap.  This order of choices is essential; we require no smoothing
estimate uniform as the sphere tends to infinity.

The positive-Gauss metric on $\mathcal C\simeq\Sph^2$ has a smooth
strictly convex Euclidean isometric embedding by the smooth Weyl
embedding theorem \citep{Nirenberg1953}.  This is precisely the
spherical positive-curvature version also stated in the introduction
of \citet{ShiTam2002}.  Denote its outward Euclidean mean curvature
by $H_E$.  We will construct the Shi--Tam scalar-flat extension with
the smooth positive prescribed boundary curvature
$H_o=\sqrt{H_i^2-k_T^2}$.  The required estimate is
\begin{equation}
 m_o\le\frac1{8\pi}\int_{\mathcal C}(H_E-H_o)\,\dd A.
 \label{eq:stress-shitam-mass-bound}
\end{equation}
Before proving this extension assertion, we evaluate its right side
and record the exact coefficient of the charge term.

Scale the induced metric on $\mathcal C$ by $r^{-2}$.
Its area is $4\pi+O(r^{-3})$ and its Gauss curvature is
$1+O(r^{-3})$, with smooth angular control.  For its convex Euclidean
embedding choose an interior point as origin and use Gauss-map
coordinates $\omega$.  Let $l(\omega)>0$ be the support function.
Then the embedding is $l\omega+\nabla_\sigma l$, its inverse shape
operator is $\nabla_\sigma^2l+l\sigma$, and
\begin{equation}
 \int H_E^{\mathrm{scaled}}\,\dd A
   =\int_{\Sph^2}\tr_\sigma(\nabla_\sigma^2l+l\sigma)\,\dd A_\sigma
   =2\int_{\Sph^2}l\,\dd A_\sigma.
 \label{eq:stress-support-mean}
\end{equation}
The scaled enclosed volume is
$\frac13\int l\,dA$, and $dA=(1+O(r^{-3}))dA_\sigma$ by the Gauss
curvature estimate.  Euclidean isoperimetry bounds this volume by
$4\pi/3+O(r^{-3})$.  Positivity of $l$ now implies
$\int l\,dA_\sigma\le4\pi+O(r^{-3})$.  Scaling back in
\eqref{eq:stress-support-mean} gives
\begin{equation}
 \int_{\mathcal C}H_E\,\dd A\le8\pi r+O(r^{-2}).
 \label{eq:stress-Euclidean-H-bound}
\end{equation}
This uses no quantitative stability theorem for the Weyl embedding.

Before smoothing the stress,
\eqref{eq:stress-outer-H-expansion} and
$dA=\phi^4r^2dA_\sigma$ give
\[
 \int_{\mathcal C}H_o\,\dd A
 =8\pi r+\int_{\Sph^2}(-16\alpha+e_0^{3/2})\,\dd A_\sigma+o(1).
\]
The smoothing error in this equality can be made arbitrarily small
after fixing $r$.  Moreover, direct substitution of
$g-b=(4\alpha r^{-3}+O(r^{-3-\xi}))b$ in the mass flux gives
\begin{equation}
 p_0(g)=8\langle\alpha\rangle.
 \label{eq:stress-conformal-mass}
\end{equation}
For clarity, if $e=fb$, the flux one-form is
$-2(V_0\,df-f\,dV_0)$; for $f=4\alpha r^{-3}$ its radial sphere
integral tends to $32\int\alpha\,dA_\sigma$.
Consequently \eqref{eq:stress-Euclidean-H-bound} gives
\begin{align}
 \frac1{8\pi}\int_{\mathcal C}(H_E-H_o)\,\dd A
 &\le p_0(g)-\tfrac12\langle e_0^{3/2}\rangle+o(1)\notag\\
 &\le p_0(g)-\tfrac12\langle e_0\rangle^{3/2}+o(1)
 \le p_0(g)-\tfrac12a^3+o(1).
 \label{eq:stress-cubic-budget}
\end{align}
The second inequality is Jensen's inequality.  If $a=0$, its charge
term is simply nonnegative.  In all cases first choosing $M_e$,
then $r$, then the compact smoothing tolerance makes the final error
smaller than the prescribed $\varepsilon_0$.

\emph{The scalar-flat extension and its mass.}
We include the extension argument to fix both the existence hypotheses
and the normalization in \eqref{eq:stress-shitam-mass-bound}.  It is
the quasi-spherical construction of \citet[Section~2, Theorem~2.1,
and Lemma~4.2]{ShiTam2002}.  The input here is exactly a smooth
strictly convex Euclidean sphere and a smooth positive prescribed
mean curvature; no compact fill-in is used.

Let $t\ge0$ be Euclidean distance along the outward normals to the
convex embedding.  The parallels foliate its entire exterior.  Write
the Euclidean metric as $dt^2+\gamma_t$, and denote the Euclidean
mean curvature and surface scalar curvature by $H_E(t)$ and $R_t$.
Both are positive.  If $\lambda_1,\lambda_2>0$ are the initial
principal curvatures, then those at time $t$ are
$\lambda_j/(1+t\lambda_j)$, whence
\begin{equation}
 \frac{R_t}{2H_E(t)}
 =\frac1{2t+c(\omega)},\qquad
 c=\lambda_1^{-1}+\lambda_2^{-1}>0.
 \label{eq:stress-parallel-curvature}
\end{equation}
Solve the scalar parabolic equation
\begin{equation}
 2H_E\partial_tU
 =2U^2\Delta_{\gamma_t}U+(U-U^3)R_t,
 \qquad U(0,\cdot)=H_E(0,\cdot)/H_o.
 \label{eq:stress-quasispherical-equation}
\end{equation}
Smooth positive initial data give a local solution by strictly
parabolic theory.  The reaction term is nonpositive above one and
nonnegative below one.  The maximum principle therefore keeps $U$
between fixed positive constants.  On every finite time interval
the equation is uniformly parabolic with smooth background geometry.
Scalar parabolic H\"older estimates for bounded solutions, followed
by Schauder estimates and bootstrapping, give continuation for all
finite times.

For the asymptotic estimate, choose positive constants respectively
above and below the entire initial range and one.  Solve, for each
of these constants as initial value,
\begin{equation}
 \dot z=(z-z^3)/(2t+c_{\max}),\qquad
 c_{\max}=\max c.
 \label{eq:stress-parabolic-barriers}
\end{equation}
By \eqref{eq:stress-parallel-curvature} these are respectively upper
and lower barriers: the sign of $z-z^3$ reverses the inequality in
the upper case.  Since
\[
 z^{-2}-1=(z(0)^{-2}-1)\frac{c_{\max}}{2t+c_{\max}},
\]
comparison gives $U-1=O((1+t)^{-1})$.
Put $s=\log(1+t)$ and $\widehat\gamma_t=(1+t)^{-2}\gamma_t$.
In the Gauss-map parametrization the parallel embedding is
$X(\omega)+t\omega$, so these scaled metrics and all their
derivatives have bounded geometry.  Equation
\eqref{eq:stress-quasispherical-equation} in $s$ is uniformly
parabolic.  Its local H\"older and then Schauder estimates on fixed
$s$-strips first bound all derivatives of $U$ uniformly.  Next apply
the linear estimates to $U-1$ with the actual coefficients of this
solution; its equation is homogeneous, with a bounded smooth
zeroth-order coefficient.  The supremum bound just proved gives
the same $O(e^{-s})$ decay at every fixed derivative order.  This
proves the stated ordinary AF derivative bounds in the Euclidean
coordinates $x'=X(\omega)+t\omega$.

Set
\begin{equation}
 g_o=U^2dt^2+\gamma_t.
 \label{eq:stress-af-extension-metric}
\end{equation}
It has the required induced metric and inner mean curvature
$H_E/U=H_o$.  Its scalar curvature is, by the hypersurface Gauss
and mean-variation formulas,
\begin{equation}
 R_{g_o}=R_t(1-U^{-2})+2H_EU^{-3}\partial_tU
                    -2U^{-1}\Delta_{\gamma_t}U=0,
 \label{eq:stress-af-scalar-zero}
\end{equation}
where the final equality is exactly
\eqref{eq:stress-quasispherical-equation}.  The positive bounds on
$U$ and the Euclidean completeness of the parallel exterior give
completeness with the boundary included.  There is precisely one
end, and $g_o-\delta=(U^2-1)dt^2$ has the stated decay.

Define
\[
 \mathcal M(t)=\int_{\mathcal C_t}H_E(1-U^{-1})\,\dd A_{\gamma_t}.
\]
The Euclidean identities
$\partial_tH_E=-|\mathrm{II}_E|^2$,
$\partial_t dA=H_EdA$, and
$H_E^2-|\mathrm{II}_E|^2=R_t$, together with
\eqref{eq:stress-quasispherical-equation}, give
\begin{align}
 \mathcal M'(t)
 &=\int_{\mathcal C_t}\left[
       R_t(1-U^{-1})+H_EU^{-2}\partial_tU\right]\,\dd A\notag\\
 &=-\frac12\int_{\mathcal C_t}R_t(U+U^{-1}-2)\,\dd A\le0.
 \label{eq:stress-shitam-monotonicity}
\end{align}
Here the integral of $\Delta_{\gamma_t}U$ vanishes.  The asymptotic
bounds show that $\mathcal M$ is bounded and that its derivative
is integrable at infinity, so it has a finite limit.

We identify the limit with the ADM mass in the normalization
\eqref{eq:adm-mass}.  For the purely normal Euclidean metric error
$e'=(U^2-1)dt^2$, the ADM flux through the parallel surface is exactly
\[
 \int_{\mathcal C_t}
   (\divg_\delta e'-d\tr_\delta e')(\partial_t)\,\dd A
 =\int_{\mathcal C_t}H_E(U^2-1)\,\dd A.
\]
The normal derivatives of $U^2-1$ cancel, and
$\nabla_{\partial_t}\partial_t=0$ in the Euclidean metric, leaving
only the tangential divergence $H_E$.  The scalar-curvature
linearization and $R_{g_o}=0$ show that the divergence of this
Euclidean flux is $O(|x'|^{-4})$.  Its integral between a large
parallel and a comparable large round sphere tends to zero; thus
it computes the ordinary ADM mass.  Finally
\[
 (U^2-1)-2(1-U^{-1})=O((U-1)^2)
\]
and $H_E\,\Area(\mathcal C_t)=O(t)$, so
\begin{equation}
 16\pi m_o
 =\lim_{t\to\infty}\int_{\mathcal C_t}H_E(U^2-1)\,\dd A
 =2\lim_{t\to\infty}\mathcal M(t)
 \le2\mathcal M(0).
 \label{eq:stress-adm-identification}
\end{equation}
As $H_E/U(0)=H_o$, this proves
\eqref{eq:stress-shitam-mass-bound}.  Combining it with
\eqref{eq:stress-cubic-budget} gives
\eqref{eq:stress-attachment-budget}.

Lastly, convexity gives
$\gamma_t(V,V)=\gamma_0((\Id+t\mathrm{II}_E^{\sharp})V,
(\Id+t\mathrm{II}_E^{\sharp})V)\ge\gamma_0(V,V)$.
The metric in \eqref{eq:stress-af-extension-metric} has no mixed
normal--tangential terms.  Its projection along the parallels to
$(\mathcal C,\gamma_0)$ therefore has operator norm at most one,
and hence upper two-dimensional Jacobian at most one.  This is
the area comparison required in the later transfer argument.
\end{proof}

All data in Proposition~\ref{prop:af-attachment} are now fixed before
the compact graph parameters are chosen.  On $S$ and $\mathcal C$
we use the base normals directed toward the end; derivatives and
fluxes on the two sides of $\mathcal C$ use $g$ and $g_o$ respectively.
Their boundary area forms agree.  In the next section the notation
$\tau=\tr_gK$ refers to the synthetic trace, not to the original
asymptotic decay exponent.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/06-stress-extension.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/07-compact-construction.tex ====
\section{The compact graph construction}\label{sec:compact-construction}

Fix the compact data and the asymptotically flat attachment supplied by
Proposition~\ref{prop:af-attachment}. Thus the connected compact inner
manifold $(\Omega_i,g)$ has boundary $S\sqcup\mathcal C$, where
$\mathcal C$ is a sphere, and carries a smooth negative definite tensor
$K$. Put $\tau=\tr_g K$. The smooth exterior $(\Omega_o,g_o)$ is scalar
flat and asymptotically flat. The induced metrics on $\mathcal C$ agree.
All normals $n$ at $S$ and $\mathcal C$ point towards the designated end;
in particular the outward normal of $\Omega_i$ at $S$ is $-n$. Write
$k_T=\tr_{T\mathcal C}K$ on the seam and use the analogous notation on
$S$. The fixed data satisfy
\begin{equation}\label{eq:compact-base-data}
 \begin{gathered}
 H_g(S)\le0,\qquad H_i>|k_T|,\qquad
 H_o=\sqrt{H_i^2-k_T^2},\\
 D(v):=R_g+\tau^2-|K|^2-2(\divg_gK-\dd\tau)(v)>0
 \quad (|v|_g\le1).
 \end{gathered}
\end{equation}
No deformation parameter changes these base data. In this section every
integral without a metric subscript uses the base metric on its own side.
Piecewise integrals mean the sum of the two such integrals. Constants may
depend on these fixed smooth data.

On $\Omega_i$ define the fields, for $N_i>0$, by
\begin{equation}\label{eq:compact-constitutive}
 \begin{gathered}
 p=s^2\nabla f,\qquad s^2+|\dd f|^2s^4=N_i^2,\qquad v=p/N_i,\\
 z=|v|,\quad y=z^2,\quad d=\sqrt{1-y},\quad W=d^{-1}=N_i/s.
 \end{gathered}
\end{equation}
The positive root specifies $s$ smoothly, including at $\dd f=0$.
On a constant-height boundary we write $\widehat z=v\cdot n$.
On $\Omega_o$ put $f=p=0$, $s=N_o$, $W=1$, and $\bar g=g_o$;
inside put $\bar g=g+s^2\dd f^2$. On either side set
$x=sw$, $h=x\bar g$, and $A=Ns\bar g^{-1}$, with $N=N_i$ or $N_o$.
Here $d$ is a scalar, whereas $\dd$ denotes differentiation.

For a finite height $M\ge1$ and a homotopy parameter $u_*\in[0,1]$ the
equations are
\begin{equation}\label{eq:compact-system}
\begin{aligned}
 \divg_g p&=TN_i &&\text{on }\Omega_i,\\
 B&=\nabla N_i-(u_*K+\ell g)p,\qquad
 \ell=T-u_*\tau &&\text{on }\Omega_i,\\
 B&=\nabla N_o,\qquad \Sigma_-=0 &&\text{on }\Omega_o,\\
 \divg B&=\Sigma_++\Sigma_-,\qquad
 -\divg(A\dd w)=\Sigma_+x+P &&\text{on each side}.
\end{aligned}
\end{equation}
We first truncate the exterior at a smooth distant coordinate sphere
$\mathcal S_R$. The complete boundary conditions are
\begin{equation}\label{eq:compact-boundary}
\begin{array}{c|l}
 S&f=u_*M,\quad B_n=-b_*,\quad (A\dd w)_n=0,\\
 \mathcal C&f=0,\quad N_o=s_i,\quad (B_i)_n=(B_o)_n,
       \quad w_i=w_o,\quad (A_i\dd w_i)_n=(A_o\dd w_o)_n,\\
 \mathcal S_R&N_o=w=1.
\end{array}
\end{equation}
Here $b_*>0$ is fixed throughout the homotopy. Eventually $R\to\infty$
with every other parameter fixed, and the last row becomes $N_o,w\to1$.

We specify all cutoffs used in these equations. Let $d_S$ be a fixed
smooth function, positive away from $S$ and equal to inward distance on
a collar of $S$. Set $F=u_*M-f$. The quotient $F/d_S$ at $S$ is its
continuous extension. For example, in that collar it equals
$\int_0^1\partial_nF(q,td_S)\,\dd t$; hence it has one fewer controlled
derivative than $F$ in the relevant H\"older spaces. Choose smooth
cutoffs whose product $\chi_T$ is supported where
\begin{equation}\label{eq:compact-trace-cutoff}
 F<M/10,\qquad z<1/2,\qquad FN_i\sqrt{x}<M^{-1},\qquad
 N_iF/d_S<M^{-1},
\end{equation}
and equals one if all four upper bounds hold with an additional factor
$1/2$. The slope cutoff is a smooth function of $z^2$. The other scalar
cutoffs are extended as one for sufficiently negative arguments. Define
\begin{equation}\label{eq:compact-sources}
 T=u_*\tau(1-\chi_T),\qquad
 \Sigma_-=-\tfrac12N_i\ell^2\zeta(N_i),\qquad
 \Sigma_+=u_*LW\eta_M\chi_x(x)\chi_N(N/\eps).
\end{equation}
The fixed smooth function $\zeta$ lies in $[0,1]$, equals one for
$N\le\delta_0$, and vanishes for $N\ge2\delta_0$.
Here $0<\delta_0,b_*,\delta<1$, $L\ge1$, $0<\eps<1$;
$\chi_x=1$ on $x\le1-\delta$ and is zero on $x\ge1-\delta/2$,
whereas $\chi_N=0$ on $N\le\eps$ and is one on $N\ge2\eps$.
In particular
\begin{equation}\label{eq:compact-source-size}
 u_*\tau\le T\le0,\quad |\ell|\le C,\quad
 -C N_i\le\Sigma_-\le0,\quad
 \|\Sigma_-\|_{L^1}\le C\delta_0,\quad 0\le\Sigma_+\le LW.
\end{equation}

The spatial cutoff $\eta_M$ is chosen as follows. Let $\psi_\infty$
be the exterior capacitary potential, equal to one on $\mathcal C$ and
zero at infinity. Fix a large radius $R_\delta$ beyond which
$\psi_\infty<\delta$. Such a potential and radius follow from scalar
Dirichlet theory and the barriers $C|x'|^{-\beta}$, $0<\beta<1$,
on the fixed asymptotically flat end. Fix an exterior collar width
$a_o>0$. The designated set $\mathcal D_M$ consists of all of
$\Omega_i$ except an inner seam collar of width $2M^{-4}$, together
with the exterior between collar distance $2a_o$ and radius
$R_\delta$. Enlarge $R_\delta$ if necessary to contain that collar.
Take $\eta_M=1$ on a neighborhood of $\mathcal D_M$, supported away
from the seam, with inner separation at least $M^{-4}$ and exterior
separation at least $a_o$, and with support in a common compact region.
All spatial cutoffs below have slightly nested supports of this kind.
Thus $N_o$ is harmonic on the fixed exterior seam collar. The choice of
$R_\delta$ does not involve an unknown solution: the exterior equations
give $N_o\Delta_{g_o}x=-P\le0$, so comparison gives
\begin{equation}\label{eq:compact-capacity-comparison}
 x\ge1-\psi_\infty \quad\text{on }\Omega_o.
\end{equation}
The same comparison holds on truncations since the right side is at
most one on $\mathcal S_R$. Consequently every exterior point with
$x<1-\delta$ lies inside the fixed radius $R_\delta$.

There are two further sources, $P=u_*(P_1+P_2)$. Put
$T_1=M^{9/10}$ and $T_2=C_2/\eps$. The source $P_1$ is an adjustable
finite constant $D_1$ times smooth cutoffs, supported inside where
$f>M/2$ and $w<4T_1$, and equal to $D_1$ where
$f\ge4M/5$ and $w\le2T_1$. The source $P_2$ is $D_2$ times smooth
cutoffs on the two spatial supports just described, vanishes if
$N\ge4\eps$ or $w\ge4T_2$, and equals $D_2$ on a neighborhood of
$\mathcal D_M$ where $N\le3\eps$ and $w\le2T_2$.
The constants $C_2,D_1,D_2$ will be chosen below. Every equation is a
smooth finite equation on the open domain $N_i,N_o,w>0$.

\begin{proposition}[Finite compact construction]\label{prop:compact-solutions}
Fix base data satisfying \eqref{eq:compact-base-data}. Fix
$b_*,\delta_0,\delta,a_o>0$ and $L\ge1$. For every sufficiently large
finite $M$, $C_2$ can be chosen depending on these parameters, and then
$\eps>0$ can be chosen arbitrarily small and $D_1,D_2$ sufficiently
large but finite, so that \eqref{eq:compact-system}--\eqref{eq:compact-boundary}
at $u_*=1$ has a positive solution on the full attached exterior.
It is smooth on each closed side of $\mathcal C$ and tends to the
specified values at infinity. It satisfies
\begin{equation}\label{eq:compact-bounds}
 \begin{gathered}
 0\le f\le M,\qquad N_i,N_o\le C(L)M,\qquad
 \int_{\Omega_i}W^2\le C(L)M^2,\\
 N_i\ge c\eps,\qquad s_i,N_o\ge c(M,L)\eps.
 \end{gathered}
\end{equation}
The constants in these bounds do not depend on $\eps,D_1,D_2$ or the
outer truncation. Dependence on the fixed data and on the preceding
parameters is suppressed. At fixed $M,L,\eps$, the lapse and height
have uniform local continuity bounds off the seam, including at $S$,
independent of $D_1,D_2$. Moreover
\begin{equation}\label{eq:compact-penalty-targets}
 w\ge M^{9/10}\quad\text{where }f\ge9M/10,
 \qquad x\ge1\quad\text{on }\mathcal D_M\cap\{N\le2\eps\},
\end{equation}
and
\begin{equation}\label{eq:compact-penalty-cost}
 \int P_1\le C(L)M^{-1/10},\qquad
 \int P_2\le\omega_M(\eps),\qquad \omega_M(\eps)\longrightarrow0
 \quad(\eps\downarrow0).
\end{equation}
The function $\omega_M$ is uniform in the finite strengths and outer
radius. In particular
\begin{equation}\label{eq:compact-bad-volume}
 \int_{\mathcal D_M\cap\{x<1-\delta\}}W\le C/L,
\end{equation}
where $C$ is independent of $L,M,\eps,D_1,D_2$. On the smooth sides the
metric $h=x\bar g$ obeys the scalar and boundary identities
\eqref{eq:compact-scalar} and \eqref{eq:compact-mean-curvature} below.
Its asymptotically flat mass satisfies
\begin{equation}\label{eq:compact-mass-budget}
 m_h\le m_o+C(b_*+\delta_0)+C(L)M^{-1/10}+C\omega_M(\eps).
\end{equation}
Thus the parameter order is: fix the base data; choose $b_*,\delta_0$
for the mass tolerance; choose $\delta,a_o$ and then $L$ for the area
tolerance; choose $M$ large; choose $C_2$; choose $\eps$ small; choose
the finite strengths; and finally let the outer radius tend to infinity.
\end{proposition}

We prove estimates for positive zeros in the H\"older domain used for
continuation in Section~\ref{sec:transmission-analysis}, before invoking
existence.  Thus $f\in C^{3,\alpha'}(\overline\Omega_i)$ and
$N,w\in C^{2,\alpha'}$ on each closed side, for the fixed positive
exponent $\alpha'$ selected there.  This is membership in the domain,
not an a priori bound on its norm.  No a priori derivative bound on a
cutoff is used in the following finite estimates.

\begin{lemma}[Lapse and slope bounds]\label{lem:compact-lapse}
Every such positive zero, for $0\le u_*\le1$ and sufficiently large
outer radius, satisfies \eqref{eq:compact-bounds}, with $M$ replacing
the upper height bound by $u_*M$. These estimates are independent of
the strengths of $P_1,P_2$ and of all derivatives of $T$.
\end{lemma}

\begin{proof}
The minimum principle for the height equation gives $f\ge0$ because
$T\le0$. A maximum exceeding $u_*M$ has $F<0$ and $z=0$, so every
trace cutoff is fully active in its neighborhood and $T=0$ there.
The strong maximum principle, applied on the exceeding set, excludes
such a maximum. Hence $0\le f\le u_*M$.

Let $\psi_R$ be harmonic on the truncated exterior, one on
$\mathcal C$ and zero on $\mathcal S_R$, and set
$\gamma_R=-\partial_n\psi_R|_{\mathcal C}$. The functions $\gamma_R$
have uniform positive upper and lower bounds. Also $\psi_R$ has a
uniform positive lower bound on all exterior source supports. To check
uniformity, compare $\psi_R$ from below with a fixed finite annular
capacitary solution and from above with $\psi_\infty$, and apply
boundary estimates and the Hopf lemma on a fixed collar. Green's
formula on the exterior and the divergence theorem on the inside give
\begin{equation}\label{eq:compact-capacity}
 \int_{\Omega_i}\Sigma_+
 +\int_{\Omega_o}\psi_R\Sigma_+
 +\int_{\mathcal C}\gamma_Rs_i
 =\int_{\mathcal C}\gamma_R+\int_S b_*
      -\int_{\Omega_i}\Sigma_-\le C.
\end{equation}
For example the seam contribution in the exterior Green formula is
$-B_n-\gamma_RN_o$, and its far contribution is
$-\partial_n\psi_R$; the harmonic flux identity for $\psi_R$ gives
the displayed signs. The constant uses only
$\int|\Sigma_-|\le C\delta_0$. In particular $\min_{\mathcal C}s_i\le C$.

Testing the height equation with $f$ and using its two constant traces
gives, since $p\cdot\dd f=W^2-1$,
\begin{equation}\label{eq:compact-height-energy}
 \int_{\Omega_i}(W^2-1)
 =-u_*M\int_S p_n-\int_{\Omega_i}fTN_i
 \le CM\sup_{\Omega_i}N_i.
\end{equation}

For the lower lapse estimate freeze
$Z=(u_*K+\ell g)v$, which satisfies $|Z|\le C$, and freeze the positive
seam function $d=s_i/N_i\le1$. Solve the homogeneous density problem
with current $\nabla\widehat N_i-Z\widehat N_i$ inside, ordinary
gradient outside, divergence zero, zero current flux at $S$, matching
current at the seam, $\widehat N_o=d\widehat N_i$, and outer value one.
This is an independent linear problem. Its scalar Fredholm assertion
is proved in Lemma~\ref{lem:transmission-fredholm}, using only frozen
linear estimates, without the present nonlinear estimates. Its adjoint
has operator $\Delta+Z\cdot\nabla$ inside, ordinary Laplacian outside,
ordinary Neumann condition at $S$, continuous value at $\mathcal C$,
and $\partial_n h_i=d\partial_n h_o$ there. The maximum and Hopf
principles give zero adjoint kernel, hence invertibility. Dual testing
with nonpositive scalar forcing gives $\widehat N\ge0$. Harnack and
the Hopf lemma give strict positivity, also at the seam: a zero there
would be a zero on both sides and the opposite Hopf derivative signs
contradict current matching. The H\"older domain gives
$d\in C^{2,\alpha'}$ and $Z\in C^{1,\alpha'}$ (in fact
$Z\in C^{2,\alpha'}$), so the independent scalar lemma applies
separately to each frozen problem. Its inverse may depend on these
coefficient norms. The uniform estimate below uses only $|Z|\le C$
and $0<d\le1$ and does not use those inverse norms.

Put $H_*:=\sup\widehat N_i$. The homogeneous version of
\eqref{eq:compact-capacity} implies $H_*\ge1$, and exterior comparison
gives $\widehat N_o\le H_*$. We claim, uniformly in the frozen fields,
\begin{equation}\label{eq:compact-homogeneous-harnack}
 \inf_{\Omega_i}\widehat N_i\ge cH_*.
\end{equation}
Here it is essential that no lower bound on $d$ is required. In Fermi
coordinates on the two sides reflect the normalized exterior field
into the inside and add it to $\widehat N_i/H_*$. The two coefficient
matrices, including volume densities, agree on the face and differ by
$O(|t|)$ at distance $|t|$ from it. For the positive exterior harmonic
field, interior gradient estimates and Harnack on balls of radius
comparable to $|t|$ give
$|t|\,|\nabla\widehat N_o|\le C\widehat N_o$.
Consequently the sum $U$ satisfies a uniformly elliptic divergence
equation with flux correction bounded by $CU$ and zero natural flux
on the face. Expressing that correction as a bounded vector times $U$
and reflecting again gives the ordinary divergence drift Harnack
inequality. Harnack chains on the fixed inner region, together with
this collar estimate and normalization by the inner supremum, give a
uniform positive lower bound for $U$ on the seam. The inner trace is
at least $U/2$, because $\widehat N_o=d\widehat N_i\le\widehat N_i$.

To carry this trace bound uniformly into a collar, take a smooth small
half-domain of radius $h$ in an inner chart and replace the normalized
inner field by its harmonic function with the same boundary data.
The replacement error is at most $Ch$: the divergence forcing
$Z\widehat N_i/H_*$ is bounded, and scaled zero-Dirichlet
$W^{1,p}$ estimates with $p>3$ give precisely that bound. The harmonic
replacement has a fixed positive lower bound centrally up to its
flat face, by the lower face data and nonnegative remaining data.
First choose $h$ small enough and then use interior Harnack chains.
This proves \eqref{eq:compact-homogeneous-harnack}.

The ratio $r_N=N/\widehat N$ is continuous across the seam. On a
sublevel $r_N<\eps/H_*$ we have $N<\eps$, so $\Sigma_+=0$ and
$\Sigma_-\le0$. Dividing the density equation by $\widehat N$ gives
a scalar supersolution inequality with no zeroth order term.
At the seam its normal derivatives match with positive weights
$\widehat N_i,\widehat N_o$; the flux $-b_*$ at $S$ has the favorable
sign for a minimum. The minimum and boundary point principles exclude
a minimum below $\eps/H_*$. Therefore
$N_i\ge(\eps/H_*)\widehat N_i\ge c\eps$.

Next suppose, with $L$ fixed, that the upper estimate fails. Along a
sequence write $N_*:=\sup N_i$ with $N_*/M\to\infty$. The exterior
maximum is bounded by $\max(N_*,1)$ since $\Delta N_o\ge0$.
By \eqref{eq:compact-height-energy},
\[
 \|\Sigma_+/N_*\|_{L^2(\Omega_i)}
 \le CL\bigl(N_*^{-2}+M/N_*\bigr)^{1/2}\longrightarrow0.
\]
The normalized negative source also tends to zero, and the exterior
source tends uniformly to zero. Equation~\eqref{eq:compact-capacity}
shows that $N_o/N_*$ has seam trace tending to zero in $L^1$.
The normalized exterior field tends to zero on every open exterior
compact set. Indeed it is bounded above by $N_*^{-1}$ plus the positive
harmonic extension, decaying at infinity, of its nonnegative seam
trace. That extension is bounded by $\psi_\infty$ and tends locally
to zero off the seam by the $L^1$ trace convergence, using a fixed
annular Poisson estimate and then the decay of $\psi_\infty$.

The reflected sum of the actual normalized lapses is uniformly
H\"older continuous up to the seam by the preceding reflected-sum
argument, now with scalar forcing tending to zero in $L^2$.
Interior Laplace divergence estimates give the same compactness away
from the seam and at $S$. After passage to a subsequence the open
inner limit $V$ extends continuously to the seam using the limit of
the sum. It is nonnegative and nonzero: an inner supremum equal to one
either lies away from the seam or gives a sum at least one in the
seam collar. With $Z$ converging weakly star, this limit satisfies the
homogeneous density equation with natural zero flux on both inner
faces. Here is a way to justify the latter assertion without individual
inner trace compactness. Use inner smooth test functions with ordinary
zero normal derivative at both faces, extended across the seam with
matching values and outer support in the harmonic collar. Integrate
the lapse derivatives onto the tests. All outer volume and trace
terms vanish by the convergence just proved, as do the normalized
sources and $b_*$. Thus
\[
 \int_{\Omega_i}\bigl(V\Delta\varphi+VZ_\infty\cdot\nabla\varphi\bigr)=0
 \quad\text{for every smooth ordinary Neumann test }\varphi.
\]
Solve the Neumann Laplace problem with divergence datum $VZ_\infty$.
Testing the difference by Neumann Poisson solutions shows that its
solution differs from $V$ by a constant. Hence $V\in H^1$ and the
natural zero-flux equation holds in the usual weak sense. Reflection
and Harnack show $V>0$ on the whole closed inner region. In particular
the original normalized inner fields are bounded below away from the
seam; near it the sum, the trace inequality, and the harmonic
replacement argument above give the same lower bound. The replacement
error now has the additional vanishing scaled $L^2$ forcing term.

We reach a contradiction by the differential estimate of
Lemma~\ref{lem:gradient-maximum}. Put $G=-\log s$ and maximize
$G+j(f)$ on $\Omega_i$, where
$j(f)=D(f^2-2Mf)$ and $D>0$ is fixed sufficiently large depending on
$L$. Since $j\le0$ on the height range and $j(0)=0$,
at the maximum $s\le\min_{\mathcal C}s\le C$. Since
$N_i\ge cN_*$ there, $y\to1$. At an interior maximum that lemma gives
\begin{equation}\label{eq:compact-gradient-test}
 j''q|\dd f|^2\le C\{1+\Sigma_+/N_i+
                    (1-y)^2(j'|\dd f|)^2\},\qquad
 q=\frac{1-y}{1+y}.
\end{equation}
Use $q|\dd f|^2=J/s^2$, $J=y/(1+y)$,
$\Sigma_+/N_i\le L/s$, and
$(1-y)^2(j'|\dd f|)^2\le(j'/N_i)^2=o(1)$.
Multiplication by $s^2$ contradicts \eqref{eq:compact-gradient-test}
once $D$ exceeds a fixed multiple of $1+L$ and the fixed upper bounds
for $s^2$ and $s$ at this maximum.

At a constant-height boundary, write $\widehat z=v\cdot n\le0$.
Lemma~\ref{lem:graph-boundary} gives
\begin{equation}\label{eq:compact-boundary-gradient}
 d^2\partial_n(G+j(f))
 =-B_n/N_i+u_*k_T\widehat z-yH+j'\widehat z/N_i.
\end{equation}
At $S$ this is strictly positive because $j'\le0$, $H\le0$,
$k_T<0$, and $B_n=-b_*$. That is impossible for an inward derivative
at a maximum. At a seam maximum $s$ is its seam minimum. The positive
harmonic exterior lapse on the fixed collar satisfies
$B_n\ge-Cs$ at that point: compare with the seam minimum times a
fixed harmonic function equal to one on the seam and zero on the
other collar boundary. Equation~\eqref{eq:compact-boundary-gradient}
is then at most $Cd+|k_T|z-yH_i+o(1)<0$, using
$H_i>|k_T|$. That is impossible for an outward derivative at a maximum.
This proves $N_i,N_o\le C(L)M$ and, by
\eqref{eq:compact-height-energy}, the stated $L^2$ estimate for $W$.

Finally keep $M,L$ fixed and suppose $\inf s_i/\eps\to0$.
Choose a fixed smooth $j$ on $[0,M]$ with $j''>0$ and
$0<j'<b_*/2$. At a maximum of $G+j(f)$, bounded oscillation of $j$
implies $s_i/\eps\to0$. The lapse floor gives $y\to1$.
Multiply \eqref{eq:compact-gradient-test} by $s^2$: its left side
stays positive, while its right side is bounded by
$C(s^2+Ls+(j')^2d^2)$ and tends to zero. At $S$, the first and last
terms of \eqref{eq:compact-boundary-gradient} have sum at least
$(b_*-j')/N_i>0$. At the seam its last term is nonpositive, and the
same harmonic collar comparison makes the derivative strictly
negative. These contradictions prove $s_i\ge c(M,L)\eps$.
On the exterior, below the level $\eps$ the positive source vanishes
and the lapse is harmonic. Its boundary values and the minimum
principle therefore give $N_o\ge c(M,L)\eps$ too.
\end{proof}

\begin{lemma}[Finite penalty selection]\label{lem:compact-penalties}
For fixed preceding parameters the sources can be selected with finite
strengths to obtain \eqref{eq:compact-penalty-targets}--\eqref{eq:compact-bad-volume}
in every actual positive solution, uniformly in the outer truncation.
Their costs obey \eqref{eq:compact-penalty-cost}.
\end{lemma}

\begin{proof}
All assertions in this proof concern $u_*=1$. First choose
$C_2\ge c(M,L)^{-1}$ from Lemma~\ref{lem:compact-lapse}, so that
$sT_2\ge1$. This precedes the choice of $\eps$. For a piecewise
Lipschitz test $\varphi$ with common seam value and zero outer value,
testing the positive $w$ equation with $\varphi^2/w$ gives
\begin{equation}\label{eq:compact-penalty-test}
 \int P\frac{\varphi^2}{w}\le\int A(\dd\varphi,\dd\varphi).
\end{equation}
Indeed the omitted term $\Sigma_+x\varphi^2/w$ is nonnegative,
and completing the square in the two derivative terms gives the
right side. Flux matching cancels the seam terms and the inner flux
is zero.

Take $\varphi=f/M$, extended by zero outside. Directly from
\eqref{eq:compact-constitutive},
$A(\dd f,\dd f)=W-W^{-1}$. On the support of $P_1$ we have
$\varphi>1/2$ and $w<4T_1$. Thus
\[
 \int P_1\le\frac{16T_1}{M^2}\int_{\Omega_i}W
 \le C(L)M^{-1/10},
\]
by Cauchy--Schwarz and \eqref{eq:compact-bounds}.

For $P_2$ take spatial localizers equal to one on its supports and
vanishing on slightly larger collars of the seam, multiplied by a
scalar cutoff of $N/\eps$ which is one below four and zero above five.
These tests may meet $S$ and have zero seam value on each side. Since
$A\le N^2g_{\rm base}^{-1}$ and $w<4T_2$ on the source support,
\begin{equation}\label{eq:compact-low-lapse-cost}
 \int P_2\le C_M\eps+
 C_M\eps^{-1}\int_{\mathcal U\cap\{N<5\eps\}}|\nabla N|^2.
\end{equation}
The fixed localization sets $\mathcal U$ stay away from the seam;
constants here may depend on $M$ and every preceding parameter.
Testing the lapse equation with a squared spatial cutoff times
$(6\eps-N)^+$ gives
\begin{equation}\label{eq:compact-low-lapse-energy}
 \int_{\mathcal U\cap\{N<5\eps\}}|\nabla N|^2
 \le C_M\eps^2+
 C\eps\int_{\mathcal U'\cap\{N<6\eps\}}\Sigma_+.
\end{equation}
To check the signs, on this sublevel the current correction has size
at most $CN\le C\eps$, so Young's inequality absorbs its gradient
term and the cutoff terms cost $C_M\eps^2$. The negative source can
be discarded. At $S$ the outward current equals $b_*>0$, so its
contribution decreases the right side of the energy inequality.

We next prove that the last integral tends uniformly to zero as
$\eps\downarrow0$. This step does not use a positive lapse floor.
If uniformity failed, take a violating sequence. Equations
\eqref{eq:compact-height-energy} and \eqref{eq:compact-bounds} bound
the positive sources in $L^2$, independently of $\eps$ and the
strengths. The lapse equation with bounded divergence datum $ZN$
therefore gives local $W^{1,p}$ bounds for every $3<p<6$, also at $S$
with the fixed flux data. After passing to a subsequence, the lapses
converge uniformly to a nonnegative function $V$ on the localization
sets, the drifts converge weakly star, and the sources converge weakly
in $L^2$. The limit equation has the form
\[
 \Delta V=\divg(Z_\infty V)+g_1.
\]
The negative-source limit vanishes almost everywhere on $\{V=0\}$
because $|\Sigma_-|\le CN$.

For completeness the scalar source $g_1$ also vanishes there. Let
$q$ be a density point of $\{V=0\}$ and a Lebesgue point of $g_1$
and $|g_1|^2$ in an interior chart. Rescale a ball of radius $r$ to a
unit ball and divide $V$ by $r^2$. The resulting nonnegative functions
vanish at the center. The inhomogeneous Harnack estimate gives a
uniform bound on a smaller ball: the rescaled drift is $rZ_\infty$,
and the rescaled scalar forcing has bounded $L^2$ norm at this
Lebesgue point. Since the zero sets occupy a fraction tending to one,
the rescaled functions tend to zero in local $L^1$. Testing their
equations against a fixed smooth bump, with derivatives transferred
to that bump, gives $g_1(q)=0$. Thus the nonnegative positive-source
limit $\sigma_+$ vanishes almost everywhere on $\{V=0\}$ as well.
Boundary points do not matter for these volume integrals. For each
$t>0$, uniform convergence implies
$\{N<6\eps\}\subset\{V\le t\}$ along the sequence eventually.
Weak $L^2$ convergence, tested against the characteristic function of
this fixed set, and then $t\downarrow0$, show
\[
 \limsup\int_{\mathcal U'\cap\{N<6\eps\}}\Sigma_+
 \le\int_{\mathcal U'\cap\{V=0\}}\sigma_+=0.
\]
Combining this with \eqref{eq:compact-low-lapse-cost} and
\eqref{eq:compact-low-lapse-energy} proves the second cost in
\eqref{eq:compact-penalty-cost}, uniformly in both strengths.

Now fix such an $\eps$. The lapse and slope bounds give uniform
ellipticity, independent of strengths. They also give uniform
continuity of $N$ and $f$ on the localization sets. For $N$, use the
same Laplace divergence estimates, now with bounded scalar forcing;
for $f$ the already established gradient bound suffices. Consequently
each point with $f\ge9M/10$ has a neighborhood of a uniform positive
radius on which $f\ge4M/5$. It is uniformly separated from the seam
unless the neighborhood ends at $S$, where the constant trace permits
reflection. Likewise every point of
$\mathcal D_M\cap\{N\le2\eps\}$ has a uniform neighborhood within
the full spatial cutoff range and with $N\le3\eps$.

In either neighborhood let $T_j$ be its corresponding threshold.
The volume of $\{w\le2T_j\}$ there is at most
$D_j^{-1}\int P_j$, hence tends uniformly to zero as $D_j\to\infty$.
The function $(2T_j-w)^+$ is a bounded weak subsolution of the
homogeneous scalar divergence equation with coefficient $A$.
Its local supremum is bounded by its $L^2$ mean, with a uniform
constant on these fixed neighborhoods; at $S$ use the homogeneous
natural-flux reflection. Since its support has the preceding small
volume and its value is at most $2T_j$, this supremum is strictly
less than $T_j$ once the strength is sufficiently large. This gives
$w\ge T_1$ on the height target and $w\ge T_2$ on the low-lapse
target. The latter implies $x\ge1$ since $sT_2\ge1$.
The choices can be made with a strict margin, so the weak inequalities
survive limits of outer truncations.

If $x<1-\delta$ at a designated point, the second target now forces
$N>2\eps$. All cutoffs in $\Sigma_+$ equal one, so
$\Sigma_+=LW$ there. Apply \eqref{eq:compact-capacity}, using the
positive lower bound of $\psi_R$ on the common exterior support.
This proves \eqref{eq:compact-bad-volume} with a constant independent
of $L,M,\eps$ and the strengths.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:compact-solutions}]
Choose the parameters in the stated order, using
Lemmas~\ref{lem:compact-lapse} and \ref{lem:compact-penalties}.
Proposition~\ref{prop:transmission-existence}, proved independently in
the next section, now supplies a positive zero for each sufficiently
large compact truncation. Its hypotheses are exactly the finite
smooth cutoffs above, the homotopy bounds of
Lemma~\ref{lem:compact-lapse}, and the fixed exterior harmonic collar.
Its continuation starts with the ordinary scalar transmission problem
at $u_*=0$; it does not assume solvability of the coupled problem.
That proposition also supplies an expanding-truncation subsequence
converging smoothly on compact subsets of each closed side, with
$N_o,w\to1$ at infinity. All the bounds and penalty conclusions pass
to this limit. This is the only invocation of nonlinear existence in
the present proof.

We record the geometric conclusions precisely. Set
$\pi=N_i^{-1}\operatorname{sym}\nabla p^\flat$,
$\lambda=\pi-K$ inside, and set these fields to zero outside.
Lemma~\ref{lem:graph-current}, applied at $u_*=1$, gives
\begin{equation}\label{eq:compact-scalar}
 xR_h=D(v)+|\lambda|^2-\ell^2-2\Sigma_-/N
       +2P/(Nx)+\tfrac32|\dd\log x|_{\bar g}^2.
\end{equation}
On the exterior take $D=R_{g_o}=0$. The current
$Q=B+A\dd w$ has matching normal flux and
$\divg Q=(1-x)\Sigma_++\Sigma_- -P$. At a constant-height face,
Lemma~\ref{lem:graph-boundary} gives
\begin{equation}\label{eq:compact-mean-curvature}
 s\sqrt{x}\,H_h
 =N(H-k_T\widehat z)+B_n+(A\dd w)_n/x.
\end{equation}
At $S$ this is strictly negative. At $\mathcal C$ the induced metrics
agree, because $s_i=N_o$ and $w$ is continuous. The seam mean
curvatures satisfy $H_{h,i}\ge H_{h,o}$ in the specified direction:
the flux terms agree and
\[
 H_i-k_T\widehat z\ge
 \sqrt{1-\widehat z^2}\sqrt{H_i^2-k_T^2},
\]
as follows by squaring, since the difference of squares is
$(H_i\widehat z-k_T)^2$ and both relevant sides are nonnegative.

Beyond the fixed source supports, both $N_o$ and $x=N_ow$ are
harmonic and tend to one. Radial barriers and scaled estimates give
$N_o-1,x-1=O_2(|x'|^{-\beta})$ for every fixed $1/2<\beta<1$.
Thus $h=xg_o$ is asymptotically flat, and the flux of $Q$ equals
the limiting flux of $\nabla x$; the remaining terms have vanishing
integral by $2\beta>1$. The linear conformal ADM variation is
$m_h-m_o=-(8\pi)^{-1}\lim\int\partial_nx$. Integration of the
current equation therefore gives the exact identity
\begin{equation}\label{eq:compact-mass}
 8\pi(m_h-m_o)
 =\int_S b_*+\int\bigl[(x-1)\Sigma_++P-\Sigma_-\bigr].
\end{equation}
Since the support of $\Sigma_+$ has $x<1$, discarding its contribution
and using the source costs proves \eqref{eq:compact-mass-budget}.
The scalar curvature decays as $O(|x'|^{-2-2\beta})$ on the end,
by \eqref{eq:compact-scalar}; this has integrable decay since
$2+2\beta>3$.
\end{proof}

The omitted collars have a controlled cost in the area comparison.
Indeed $W=1$ outside, and the inner omitted collar has volume
$O(M^{-4})$. Hence
\begin{equation}\label{eq:compact-full-bad-volume}
 \int_{\{x<1-\delta\}}W
 \le C/L+C a_o+C(L)M^{-1}.
\end{equation}
This uses \eqref{eq:compact-capacity-comparison} outside $R_\delta$
and Cauchy--Schwarz with \eqref{eq:compact-bounds} in the inner
collar. It explains why $a_o$ and $L$ are chosen before $M$.
Finally the possible negative scalar curvature is confined to the
closed set defined by the weak inequalities in
\eqref{eq:compact-trace-cutoff} and $N_i\ge\delta_0$.
Outside that set either $\ell=0$ or $\zeta=1$, and
\eqref{eq:compact-scalar} is nonnegative. On that set the height
penalty and $z\le1/2$ give
$x\ge(\sqrt3/2)\delta_0M^{9/10}$, so
$R_h\ge-C\delta_0^{-1}M^{-9/10}$ everywhere on the smooth sides.
The geometric treatment of this exceptional set is the content of
Section~\ref{sec:scalar-error}.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/07-compact-construction.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/08-transmission-analysis.tex ====
\section{Regularity and continuation at the transmission surface}
\label{sec:transmission-analysis}

We prove the existence assertion needed in Section~\ref{sec:compact-construction}.
Throughout this section all parameters, including the penalty strengths, are
finite.  Constants in estimates for the lapse and height state may depend on
the fixed parameters preceding the strengths, but not on those strengths.
Constants involving the conformal factor or higher derivatives may also
depend on the strengths.  Neither assertion requires uniform ellipticity as
the internal parameters tend to their final limits.

Write $\Omega_R=\Omega_i\cup_{\mathcal C}\Omega_{o,R}$, with far boundary
$S_R$.  On both sides of $\mathcal C$ the normal $n$ points toward the end.
Thus equality of the two $n$-fluxes is the condition for cancellation in
Green's formula.  The two base metrics have the same induced metric at
$\mathcal C$.  Joined Fermi charts have coordinates $(z',t)$, with
$t<0$ inside, $t>0$ outside, and $n=\partial_t$ at the face.  Coordinate
divergence currents below include the relevant volume density.  We retain
the notation $d=s/N$, $y=1-d^2$, $q=(1-y)/(1+y)$,
$I=(1+y)^{-1}$, $J=y/(1+y)$, and $\beta=2/(1+y)$ from the graph calculus.

\begin{proposition}[Finite-parameter transmission existence]
\label{prop:transmission-existence}
Fix the smooth compact base data and AF attachment of
Section~\ref{sec:compact-construction}, with $H_i>|k_T|$ on
$\mathcal C$, $H_g(S)\leq0$, and the smooth negative definite synthetic
tensor $K$.  Fix positive finite parameters and smooth cutoffs as in
\eqref{eq:compact-system}--\eqref{eq:compact-trace-cutoff}, including a
source-free exterior collar of $\mathcal C$ and the constant $b_*>0$.
Then on every sufficiently large smooth truncation there is a positive
piecewise smooth solution of \eqref{eq:compact-system} and
\eqref{eq:compact-boundary} at $u_*=1$.

At these fixed parameters a subsequence of such solutions as $R\to\infty$
converges smoothly on each closed side on compact subsets to a positive
piecewise smooth solution on the full joined exterior.  Its end values are
$N_o,w\to1$.  The bounds in \eqref{eq:compact-bounds} hold, and all
compact-set derivative bounds are uniform in $R$.  The penalty conclusions
in Section~\ref{sec:compact-construction}, when the strengths have been
chosen there, therefore hold for these solutions.
\end{proposition}

The proof uses the a priori estimates of Lemma~\ref{lem:compact-lapse},
which were proved for positive classical zeros without assuming existence.
The scalar linear transmission fact used in that lemma is established
independently below as part of Lemma~\ref{lem:transmission-fredholm}.
In particular there is no appeal to
Proposition~\ref{prop:compact-solutions} in proving this proposition.

\subsection{Energy and oscillation of the lapse and height state}

For the moment consider any positive homotopy zero in the continuation
domain: $f\in C^{3,\alpha'}$ on the closed inner side and
$N,w\in C^{2,\alpha'}$ on each closed side, with $\alpha'>0$.
No bound for these norms is assumed.  The exponent $\gamma$ obtained
below depends only on the finite state bounds, so the continuation
domain can subsequently be fixed with $\alpha'<\gamma$.
The finite bounds in \eqref{eq:compact-bounds} give positive constants
$c_0,C_0$ such that
\begin{equation}
 c_0\leq N_i,s_i,N_o\leq C_0,
 \qquad |\dd f|\leq C_0.
 \label{eq:transmission-state-range}
\end{equation}
Here and below an estimate for $s$ on the exterior means the same estimate
for $N_o$.  The scalar lapse sources and the drift
$(u_*K+\ell g)p$ are bounded at this stage; their derivatives are not used.

\begin{lemma}[Continuity at the nonlinear seam]
\label{lem:seam-holder}
For the state range \eqref{eq:transmission-state-range}, there are
$\gamma>0$ and $C<\infty$ such that
\begin{equation}
 \|N_i\|_{C^\gamma(\overline\Omega_i)}
 +\|\dd f\|_{C^\gamma(\overline\Omega_i)}
 +\|N_o\|_{C^\gamma(K\cap\overline\Omega_o)}\leq C
 \label{eq:transmission-holder}
\end{equation}
on every fixed compact set $K$.  The constants are uniform over homotopy
zeros and sufficiently large outer truncations.  They depend on the finite
state range, bounds for the undifferentiated height and lapse sources,
and the fixed geometry, but not on derivatives of $T$ or the strength of
$P$.
\end{lemma}

\begin{proof}
Away from $\mathcal C$, the lapse equation is a smooth metric Laplace
equation with bounded divergence forcing and bounded scalar forcing.
Its local $W^{1,p}$ bounds hold for some $p>3$, also up to $S$ with its
natural current flux.  The prescribed constant flux at $S$ can be put in
the current by a smooth bounded extension before reflection.  Differentiating
the divergence height equation gives scalar uniformly elliptic equations
for the coordinate derivatives of $f$.  Their divergence forcing contains
first derivatives of $N$, bounded in $L^p$, and the bounded scalar height
forcing, put into the corresponding component of the differentiated
current.  In particular differentiating that scalar forcing in the weak
equation does not require a bound for $\dd T$.
The inhomogeneous scalar divergence estimate gives H\"older bounds for
$\dd f$.  At $S$ subtract the constant height and reflect it oddly, the
lapse evenly, and the metric with its tensor parities in an orthogonal
collar.  The reflected metric is Lipschitz.  The height flux satisfies the
weak reflected equation, since its normal component has the required even
parity.  Difference quotients justify differentiation there.  These
arguments prove the assertion away from the seam.

At the seam we first bound energy weighted by inverse distance.
A modified Newton current then gives an estimate relative to compatible
constant states. The weighted bound supplies a uniformly positive
scale of small excess; a blowup to an explicit linear transmission
problem gives excess decay and hence H\"older continuity.

\smallskip\noindent\emph{Weighted energy.}
Set
\begin{equation}
 E=\begin{cases}|\nabla N_i|^2+|\nabla^2 f|^2,&t<0,\\
                 |\nabla N_o|^2,&t>0,
   \end{cases}
 \qquad G=-\log s,\qquad Y=s^{-k_0},
 \label{eq:transmission-energy-definition}
\end{equation}
where $k_0$ is sufficiently large for
Lemma~\ref{lem:gradient-energy}.  On the inside its divergence matrix is
$\rho\mathcal P$, where $\rho=(1+y)/d$, and on the outside it is the
base cometric.  After including the volume densities, these give a bounded
symmetric uniformly elliptic matrix $a$ in the joined chart.
$Y$ is continuous across the face because $N_o=s_i$ there.
The energy identity and the boundary identity give
\begin{equation}
 \partial_j(a^{jk}\partial_kY)
 \geq cE-C-C\delta_{\mathcal C}.
 \label{eq:transmission-joined-energy}
\end{equation}
This is a distributional inequality.  Indeed $\rho q=d$ on a
Dirichlet-height face, and the inner normal $Y$-flux is
$-k_0YB_n/s+O(1)$, whereas the outer one is $-k_0YB_n/s$.
The bounded discrepancy is precisely the surface term in
\eqref{eq:transmission-joined-energy}.

Fix concentric chart balls strictly contained in a larger joined chart.
Let $\theta_\varepsilon$ be the zero Dirichlet Green potential, for
$-\partial(a\partial)$ on the larger ball, of a nonnegative normalized
bump at a point $z_0$ in the smaller ball.  If $Y_H$ is the homogeneous
replacement of $Y$ with its boundary values, then
\begin{equation}
 \big\langle\partial(a\partial Y),\theta_\varepsilon\big\rangle
  =\int(Y_H-Y)\,\chi_\varepsilon\leq C.
 \label{eq:transmission-green-pairing}
\end{equation}
The maximum principle bounds $Y_H$, and $Y$ is bounded by
\eqref{eq:transmission-state-range}.  For each classical solution the
identity follows first with regular potentials and then by Sobolev
approximation.  The surface term is well defined by the trace theorem.
The divergence Green bounds for symmetric measurable uniformly elliptic
matrices in dimension three give an upper bound $C|z-z_0|^{-1}$ and a
comparable lower bound on a sufficiently small interior ball
\citep[Theorem~7.1 and pp.~66--67]{LSW1963}.  The surface integral of
$|z-z_0|^{-1}$ is uniformly finite.  Letting the bump shrink, using
\eqref{eq:transmission-joined-energy} and Fatou's lemma, proves
\begin{equation}
 \int_{B_{h_0}(z_0)}\frac{E(z)}{|z-z_0|}\,\dd z\leq C.
 \label{eq:transmission-green-energy}
\end{equation}
Both $h_0>0$ and $C$ are uniform on the finite collection of seam charts.

\smallskip\noindent\emph{Comparison with a constant state.}
We next prove an energy bound relative to a compatible constant state.
Such a state is
\begin{equation}
 \mathcal V_*=(n_*,a_*\dd t,s_*),\qquad
 s_*=s(n_*,|a_*|),\qquad
 \mathcal V=(N_i,\dd f,N_o).
 \label{eq:transmission-reference}
\end{equation}
The pairs $(n_*,a_*)$ range over a compact rectangle containing the
state ranges with a margin, with $n_*>0$.  Since $|\dd t|_g=1$, $s_*$ is
constant in this Fermi chart.  Norms of the state mean the sum of the
inner pair's squared norm on the inner half and the outer lapse's
squared norm on the outer half.

To derive the bound, put
$\mathcal T_f=\nabla^2f-(\Delta f)g$.  The Newton-current identity in
Lemma~\ref{lem:newton-flux} says
\begin{equation}
 \rho\mathcal P\nabla G
 =-\frac{\nabla N_i}{s}
   +\mathcal T_f\left(\frac{y\nabla f}{d|\dd f|^2}\right)+O(1).
 \label{eq:transmission-newton-current}
\end{equation}
The coefficient vector extends smoothly through $\dd f=0$.  In the
current $a\dd Y$, add $k_0Y_*B/s_*$ on both sides, where
$Y_*=s_*^{-k_0}$, and on the inside subtract the Newton term in
\eqref{eq:transmission-newton-current} with its full coefficient
$k_0Y,y\nabla f/(d|\dd f|^2)$ frozen at the reference.  Include metric
densities in all three currents.  Call the resulting joined current
$\mathcal J_*$.  The cancellation of the coefficients of $\nabla N$
and $\nabla^2f$ at the reference gives
\begin{equation}
 |\mathcal J_*|\leq C\bigl(1+|\mathcal V-\mathcal V_*|\sqrt E\bigr).
 \label{eq:transmission-modified-current-size}
\end{equation}
Moreover $\divg\mathcal T_f=\Ric(\nabla f,\cdot)$, so the divergence
of the subtracted term is bounded by $C(1+\sqrt E)$; its frozen vector
is extended smoothly using $\partial_t$.  The added $B$-current has
bounded divergence and no seam jump.  The frozen Newton vector is normal
at the seam.  Its normal contraction there involves
$-\tr_{\mathcal C}\nabla^2f$, which is bounded since $f$ is constant
along that face and $|\dd f|$ is bounded.  Young's inequality therefore
gives, after lowering $c$,
\begin{equation}
 \divg\mathcal J_*\geq cE-C-C\delta_{\mathcal C}.
 \label{eq:transmission-modified-current-divergence}
\end{equation}
Test by $\chi^2$, with $\chi=1$ on $B_h$, supported in $B_{2h}$ and
$|\dd\chi|\leq C/h$.  Absorb half the energy using
\eqref{eq:transmission-modified-current-size}.  The volume, constant
current, and surface contributions are at most $Ch^3$, $Ch^2$, and
$Ch^2$, respectively.  We obtain
\begin{equation}
 h^{-1}\int_{B_h}E\leq
 C\left(\frac{1}{|B_{2h}|}\int_{B_{2h}}
                    |\mathcal V-\mathcal V_*|^2+h\right).
 \label{eq:transmission-caccioppoli}
\end{equation}
On a pure inner ball the identical construction uses any constant
covector reference.  The coefficients frozen against this reference may
vary smoothly with the background metric; their derivatives give only
$C(1+\sqrt E)$ errors.  On a pure outer ball the analogous assertion
is the usual lapse Caccioppoli inequality.

\smallskip\noindent\emph{A scale of small excess.}
Fix $0<b_0<1$ and define the seam excess
\begin{equation}
 \mathcal E(z_0,h)=
 \inf_{\mathcal V_*}\frac{1}{|B_h|}\int_{B_h(z_0)}
          |\mathcal V-\mathcal V_*|^2+h^{b_0}.
 \label{eq:transmission-excess}
\end{equation}
There is a uniformly positive scale at which this excess is arbitrarily
small.  To see the quantitative content, put $h_j=2^{-j}h_0$.  For each
$z\ne z_0$ the sum of $h_j^{-1}$ over the indices with $|z-z_0|<h_j$
is at most $C|z-z_0|^{-1}$.  Thus
\eqref{eq:transmission-green-energy} bounds
$\sum_j h_j^{-1}\int_{B_{h_j}}E$ uniformly.  Given $\eta>0$, first
make the starting ceiling small enough that $h_0^{b_0}+h_0^2<\eta$.
Among a fixed finite number of its dyadic descendants there is a scale
with $h^{-1}\int_{B_h}E<\eta$.  The number depends only on the
Green bound and $\eta$, so the chosen scale has a uniform positive
lower bound.

Here is why this is also a small excess scale.  Scaled Poincar\'e and
the half-ball trace inequality bound both the volume deviations and the
mean-square face deviations from each side mean by
\begin{equation}
 C h^{-1}\int_{B_h}E+Ch^2.
 \label{eq:transmission-trace-poincare}
\end{equation}
The additional $Ch^2$ allows for Christoffel terms when replacing
covariant Hessians by coordinate derivatives of $\dd f$.  The
tangential trace of $\dd f$ is zero; hence the tangential components of
its mean are small.  The trace relation $N_o=s(N_i,\partial_t f)$ and
the uniform Lipschitz bound for $s$ on the state range show that the
outer mean differs from the value obtained from the two inner means by
at most the square root of \eqref{eq:transmission-trace-poincare}.
This provides a compatible reference with excess $C\eta$.  All
references obtained this way lie in a compact subrange with room for
the subsequent small updates.

\smallskip\noindent\emph{Excess decay.}
We claim that, for fixed sufficiently small $\vartheta>0$, whenever
$h$ and $\mathcal E(z_0,h)$ are sufficiently small,
\begin{equation}
 \mathcal E(z_0,\vartheta h)
 \leq 2\bigl(C\vartheta^2+\vartheta^{b_0}\bigr)
                    \mathcal E(z_0,h).
 \label{eq:transmission-excess-decay}
\end{equation}
The constant $C$ is independent of $\vartheta$.  We give the
compactness argument, including the value trace.

If the claim fails, choose violating balls with
$\sigma_j^2=\mathcal E(z_j,h_j)\to0$ and best references
$(n_j,a_j\dd t,s_j)$.  Translate and dilate these balls to unit size
using $z=z_j+h_j\zeta$, and rotate tangential coordinates to be
orthonormal at their centers.  Define the lapse differences divided by
$\sigma_j$, and the height difference by
\begin{equation}
 u_{i,j}=\frac{N_i-n_j}{\sigma_j},\qquad
 u_{o,j}=\frac{N_o-s_j}{\sigma_j},\qquad
 \phi_j(\zeta)=
       \frac{f(z_j+h_j\zeta)-a_jh_j\zeta_3}{h_j\sigma_j}.
 \label{eq:transmission-normalized-fields}
\end{equation}
The trace of $\phi_j$ is zero.  Their state has bounded $L^2$ norm on
the unit ball.  Applying \eqref{eq:transmission-caccioppoli} on smaller
balls gives locally bounded $H^1$ norms for $u_{i,j},u_{o,j}$ and
$\nabla\phi_j$.  Indeed its error after normalization is
$h_j/\sigma_j^2\leq h_j^{1-b_0}\to0$.
The $L^2$ norm of $\phi_j$ is controlled by its gradient and its zero
trace.  Rellich compactness and compactness of the $H^1$ trace into
$L^2$ on smaller face patches give strong local $L^2$ convergence of
the state in the bulk and on the face, and weak $H^1$ convergence.

Pass to a convergent reference $(n_*,a_*,s_*)$.  The derivative of
the height flux with respect to the gradient is $s_*^2\mathcal P_*$;
its lapse derivative is $s_*^2(\beta/n_*)a_*\partial_t$.
Taylor remainders divided by $\sigma_j$ have $L^1$ norm tending to
zero, since the unnormalized state difference is $\sigma_j$ times
an $L^2$-bounded function.  The same assertion holds on the face by
the trace bounds.  Smooth metric errors are $O(h_j/\sigma_j)$,
which tends to zero.  The height source and lapse drift enter weak
tests with this same factor; the scalar lapse source has the smaller
factor $O(h_j^2/\sigma_j)$.  Thus no derivative of a source is being
passed to the limit.  Joined tests have a common value on the face,
so lapse current matching passes to equality of the limiting normal
derivatives.  We arrive at
\begin{align}
 \Delta u_i&=\Delta u_o=0,
 & (q\partial_t^2+\Delta_{z'})\phi+k\partial_tu_i&=0,
 & k&=\beta a_*/n_*,\label{eq:transmission-frozen-interior}\\
 \phi&=0,
 &\partial_tu_i&=\partial_tu_o,
 &u_o&=r_0u_i+r_0c\partial_t\phi \quad(t=0),
 \label{eq:transmission-frozen-boundary}
\end{align}
where
\begin{equation}
 r_0=dI>0,\qquad c=-\frac{n_*y}{a_*},\qquad
 ck=q-1,\qquad r_0c=-\frac{s_*J}{a_*}.
 \label{eq:transmission-frozen-relations}
\end{equation}
All expressions have their continuous interpretations at $a_*=0$;
in particular $c=k=0$ there.  Differentiating
$s^2+|\dd f|^2s^4=N_i^2$ gives
$\dot s=dI\dot N_i-sJ\partial_t\dot f/a_*$, which verifies
the value relation directly.

The elementary harmonic estimate proved just below gives uniform
smooth bounds for these limits on smaller closed half-balls in terms
of their $L^2$ state norms.  Consequently their deviations from the
center values have mean square at most $C\vartheta^2$.  The center
values satisfy the linearized compatibility relation.  Replace the
old reference by
\begin{equation}
 n_j'=n_j+\sigma_j u_i(0),\qquad
 a_j'=a_j+\sigma_j\partial_t\phi(0),\qquad
 s_j'=s(n_j',|a_j'|).
 \label{eq:transmission-reference-update}
\end{equation}
The nonlinear compatibility error in the outer value is
$O(\sigma_j^2)$, and the tangential height gradient is zero at the
center.  Strong local $L^2$ convergence therefore proves
\eqref{eq:transmission-excess-decay}, contradicting its violation.
The added radius term contributes at most
$\vartheta^{b_0}\sigma_j^2$, as required.

For completeness, the harmonic estimate used here is as follows.
Reflect $u_o$ into $t<0$ and put $U=u_i+u_o^{\rm refl}$.  It is
harmonic with zero Neumann data, so it is smooth with controlled local
norms by even reflection.  On a smaller inner cylinder choose a
harmonic primitive $D$ with $D_t=u_i$.  An explicit choice is
\begin{equation}
 D(z',t)=\int_{t_0}^{t}u_i(z',v)\,\dd v+a(z'),
 \qquad \Delta_{z'}a=-\partial_tu_i(z',t_0),
 \label{eq:transmission-harmonic-primitive}
\end{equation}
where $t_0<0$ is a fixed interior slice.  The slice data are controlled
by interior harmonic estimates.  This formula gives $D\in H^2$
locally up to the face: the mixed and double normal derivatives come
from $u_i\in H^1$, and tangential Laplace estimates applied to
$\Delta_{z'}D=-\partial_tu_i$ give its remaining second derivatives.
The function $V=D+c\phi$ satisfies
$(q\partial_t^2+\Delta_{z'})V=0$, since $ck=q-1$, and $V=D$
on the face.  Set $\widetilde V(z',t)=V(z',\sqrt q\,t)$.
It is harmonic and, with $a_0=r_0/\sqrt q>0$,
\begin{equation}
 (\widetilde V-D)|_{t=0}=0,
 \qquad \partial_t(D+a_0\widetilde V)|_{t=0}=U.
 \label{eq:transmission-positive-harmonic-combinations}
\end{equation}
Scalar harmonic Dirichlet and Neumann estimates control both
combinations.  Their coefficient matrix is invertible because
$1+a_0>0$; in fact $a_0=1/\sqrt{1+y}$.  They control $D$, then
$u_i=D_t$ and $u_o=U-u_i^{\rm refl}$.  Finally the scalar Dirichlet
equation in \eqref{eq:transmission-frozen-interior} controls $\phi$.
This last step avoids division by $c$, so all estimates are uniform
through zero slope.  The normal rescaling only reduces the patch by
a uniform factor on the state range.

Choose $\vartheta$ so small, and then $\gamma>0$ so small, that
$2(C\vartheta^2+\vartheta^{b_0})\leq\vartheta^{2\gamma}$.
Begin iteration at the small-excess scales with uniform positive
lower bound constructed above.  The references stay in the permitted
range because their successive changes are bounded by a convergent
geometric sum.  This yields a mean-square oscillation bound
$Ch^{2\gamma}$ at every seam center.  On a pure ball near the seam,
the first scale comparable to its distance from the seam is controlled
by a seam ball.  The corresponding interior excess argument, with an
arbitrary constant gradient reference, then gives the bound at all
smaller scales.  Its limit equations are a harmonic lapse and a
constant uniformly elliptic height equation with the known lapse
coupling; scalar interior estimates give the same decay.  Campanato's
characterization on each closed half-chart proves
\eqref{eq:transmission-holder}.  Finitely many charts and the earlier
ordinary boundary estimates complete the proof.
\end{proof}

\subsection{Estimates with the full derivative orders}

We first record the linear estimate, including the unequal orders of
the unknowns.  This is also a proof of the relevant boundary estimate;
it does not invoke a general theorem for the nonlinear transmission
condition.

\begin{lemma}[Local transmission Schauder estimate]
\label{lem:transmission-schauder}
Give a height variation $\phi$ order $3$, lapse variations $u_i,u_o$
order $2$, and conformal-factor variations $v_i,v_o$ order $2$.
For frozen coefficients from a state in
\eqref{eq:transmission-state-range}, the principal equations are
\begin{equation}
 \mathcal P:\nabla^2\phi
       +\frac\beta N\nabla f\cdot\nabla u_i,\qquad
 \Delta u_i,\quad\Delta u_o,\qquad
 A_i:\nabla^2v_i,\quad A_o:\nabla^2v_o.
 \label{eq:transmission-principal-rows}
\end{equation}
Their residual norms are respectively $C^{1,\alpha}$ and
$C^{0,\alpha}$ for the other four rows.  The height trace has
$C^{3,\alpha}$ norm.  The lapse and conformal-factor value residuals
have $C^{2,\alpha}$ norm, and their current-flux residuals have
$C^{1,\alpha}$ norm.  On a smaller fixed patch, the field norms of
the indicated orders are bounded by these residual norms and the
supremum norms of the fields on a larger patch.  The constants are
uniform on compact positive state ranges.

At the seam the height/lapse principal conditions are
\eqref{eq:transmission-frozen-boundary}; the conformal-factor block
has value continuity and positive conormal matching.  At $S$ they
are Dirichlet for height and Neumann for the other fields; at $S_R$
they are Dirichlet.  The same estimates hold for scalar transmission
with a fixed positive $C^{2,\alpha}$ value ratio, and while its outer
normal-derivative coefficient is decreased to zero, the inner
coefficient being kept strictly positive.
\end{lemma}

\begin{proof}
It suffices first to work on a flat seam patch.  Subtract scalar
particular solutions of the two lapse Poisson equations, with
$C^{2,\alpha}$ control, and a $C^{3,\alpha}$ particular solution
of the height row, including its $C^{3,\alpha}$ boundary trace.
The height forcing includes the first derivative of the particular
inner lapse, hence belongs to $C^{1,\alpha}$ as required.  Such
particular solutions can be made on smooth domains having flat face
portions containing the smaller patch, after extending localized
data; ordinary scalar Dirichlet estimates give their stated norms.
We are left with harmonic lapses and the homogeneous height row,
zero height trace, and residuals
\begin{equation}
 \partial_tu_i-\partial_tu_o=h_1,
 \qquad u_o-r_0u_i-r_0c\partial_t\phi=h_2,
 \label{eq:transmission-inhomogeneous-seam}
\end{equation}
where $h_1\in C^{1,\alpha}$ and $h_2\in C^{2,\alpha}$.
The reflected sum $U=u_i+u_o^{\rm refl}$ is harmonic with Neumann
data $h_1$, and has a controlled $C^{2,\alpha}$ norm on a smaller
patch.  Construct $D$ by
\eqref{eq:transmission-harmonic-primitive} and use its same local
supremum bound.  The two harmonic combinations in
\eqref{eq:transmission-positive-harmonic-combinations} now have
Dirichlet datum zero and Neumann datum $U-h_2$, respectively.
The latter belongs to $C^{2,\alpha}$, so both combinations, and
therefore $D$, are controlled in $C^{3,\alpha}$.  This controls
both lapses in $C^{2,\alpha}$.  Scalar Dirichlet estimates for the
height row finish the $C^{3,\alpha}$ estimate for $\phi$.

For the conformal-factor block, the frozen inner matrix in orthonormal
face coordinates is a positive multiple of
$\operatorname{diag}(1,1,d^2)$, and the outer matrix is isotropic.
Subtract particular solutions, rescale the two normal coordinates,
and reflect the outer field.  The resulting fields are harmonic;
their difference has the prescribed value datum and a positive
weighted sum has the prescribed Neumann datum.  This gives their
$C^{2,\alpha}$ bounds.  More generally, if the value relation is
$v_o=r v_i+h$ with $r>0$, and the flux relation after reflection is
$a\partial_tv_i+b\partial_tv_o^{\rm refl}=k$, then the harmonic
combinations $v_o^{\rm refl}-rv_i$ and $av_i+bv_o^{\rm refl}$
have Dirichlet and Neumann data.  Their coefficient determinant is
$a+br>0$.  It stays positive if $b$ decreases to zero with $a>0$.
Variable fixed $C^{2,\alpha}$ ratios are treated by freezing, as in the
next paragraph.  Localization differentiates the value row at most
twice and the flux row at most once; the constants depend only on
the corresponding coefficient norms.  The ordinary boundary and interior estimates follow
from the scalar estimates, with the lapse estimated before the
height row.

For fixed regular variable coefficients, localization gives the same
estimate with lower-order terms.  In a sufficiently small chart
the leading coefficients differ little in supremum norm from their
frozen values.  Apply the constant-coefficient estimate to cutoff
fields on that chart.  In a product such as
$(a-a_*)D^2v$, the coefficient supremum multiplies the top H\"older
seminorm and can be absorbed.  Its fixed H\"older seminorm multiplies
the second-derivative supremum, which is bounded by an arbitrarily
small multiple of the full high norm plus a fixed multiple of the
field supremum, by interpolation.  The same argument applies to
the order-$1$ height coupling and to the differentiated value
relation.  Cutoff commutators have strictly lower orders.  A finite
cover and interpolation for separated-point H\"older quotients give
the claimed local or global variable-coefficient estimate.  This
argument requires only fixed coefficient bounds and is uniform
along compact families of such coefficients.
\end{proof}

We give the nonlinear application of this estimate in detail.  Fix
$0<\alpha'<\alpha<\gamma$, decreasing $\gamma$ in
Lemma~\ref{lem:seam-holder} if necessary.  On a fixed truncation use
the Banach domain
\begin{equation}
 X=C^{3,\alpha'}(\overline\Omega_i)_f
   \times C^{2,\alpha'}(\overline\Omega_i)_{N_i,w_i}
   \times C^{2,\alpha'}(\overline\Omega_{o,R})_{N_o,w_o},
 \label{eq:transmission-domain-space}
\end{equation}
with the two height Dirichlet values imposed by an affine lift of
$u_*$, and with the open condition $N_i,N_o,w_i,w_o>0$.
There are no other conditions built into this domain.  The output
space, denoted $\mathscr Y$, contains the height equation in
$C^{1,\alpha'}$, the two lapse and two conformal-factor equations
in $C^{0,\alpha'}$, the seam value residuals and the far Dirichlet
residuals in $C^{2,\alpha'}$, and the flux residuals at $S$ and
$\mathcal C$ in $C^{1,\alpha'}$.  Height boundary values are not
additional outputs because they are prescribed in $X$.

The quotient in the trace cutoff causes no additional loss.  In a
distance collar $r_S\geq0$ of $S$, with $F=u_*M-f=0$ on $S$,
\begin{equation}
 \frac{F(z',r_S)}{r_S}
   =\int_0^1\partial_{r_S}F(z',\theta r_S)\,\dd\theta.
 \label{eq:transmission-quotient}
\end{equation}
It is consequently an affine bounded map from the height domain
into $C^{2,\alpha'}$.  The same formula controls it in all the
order-by-order estimates below.  Outside this collar its denominator
is smooth and positive.  Products and the finite smooth cutoffs
therefore define a smooth residual map
\begin{equation}
 \mathscr F:[0,1]\times X_+\longrightarrow\mathscr Y,
 \label{eq:transmission-nonlinear-map}
\end{equation}
where $X_+$ denotes the positive open domain after the height lift.

At an exact zero, $w\geq1$.  Indeed its equation has nonnegative
right-hand side, its outer value is $1$, and its natural fluxes
match with zero inner flux.  Testing with $(1-w)^+$ gives the claim.
Since $\Sigma_+x$ is bounded on its support by the finite state
range and the cutoff $x<1-\delta/2$, and $P$ has finite strength,
the forcing in this scalar joined equation is bounded independently
of the particular zero.  Testing with $w-1$ and using the
Dirichlet Poincar\'e inequality gives an $H^1$ bound for $w-1$ on
the fixed truncation.  Local scalar boundedness and H\"older estimates
for its measurable uniformly elliptic joined divergence matrix give
a $C^\gamma$ bound, after reducing $\gamma>0$.  Reflection handles
the homogeneous inner flux.  Thus
\begin{equation}
 \mathscr U=(\dd f,N_i,N_o,w_i,w_o)
 \quad\hbox{is bounded piecewise in }C^\gamma.
 \label{eq:transmission-state-holder-all}
\end{equation}

We now prove an a priori $C^{2,\alpha}$ bound for $\mathscr U$.
For a smooth zero, differentiate the nondivergence height row once.
Its highest terms are the height and lapse terms in
\eqref{eq:transmission-principal-rows}.  Expanding the lapse and
$w$ equations shows that their only highest terms at these column
orders are their own second derivatives.  Every other term involves
at most two factors of first derivatives of $\mathscr U$, with
coefficients that are smooth functions of its bounded state and of
the quotient \eqref{eq:transmission-quotient}.  At the seam,
differentiating the value relation twice tangentially gives
\begin{equation}
 \partial_a\partial_b(N_o-s_i)
  =\partial_a\partial_b N_o
    -D s_i\,\partial_a\partial_b(N_i,\dd f)
    -D^2s_i\bigl(\partial_a(N_i,\dd f),
                 \partial_b(N_i,\dd f)\bigr)
    +\mathcal R_{ab},
 \label{eq:transmission-value-differentiation}
\end{equation}
where $\mathcal R_{ab}$ contains only bounded geometry coefficients
and lower-order derivatives.  The leading first differential is
exactly the principal value relation already used.  One derivative
of a current-flux relation has only the indicated first-order
principal boundary operators; its other terms are lower at these
column orders.  Formula~\eqref{eq:transmission-quotient} gives the
same derivative count for every occurrence of $F/d_S$.

Here are the interpolation exponents needed for absorption, to make
the dependence on the prior continuity estimate explicit.  Put
$H=1+\|\mathscr U\|_{C^{2,\alpha}}$ and suppose
$\|\mathscr U\|_{C^\gamma}\leq C_0$.  On fixed slightly enlarged
patches, H\"older interpolation gives
\begin{align}
 \|D\mathscr U\,D\mathscr U\|_{C^{0,\alpha}}
   &\leq C H^{\theta_1},
 &\theta_1&=\frac{2+\alpha-2\gamma}{2+\alpha-\gamma}<1,
 \label{eq:transmission-product-interpolation}\\
 \|D^2\mathscr U\|_{C^0}
   &\leq C H^{\theta_2},
 &\theta_2&=\frac{2-\gamma}{2+\alpha-\gamma}<1.
 \label{eq:transmission-sup-interpolation}
\end{align}
Terms of still lower order have smaller exponents or admit the usual
$\eta H+C_\eta$ bound.  The first estimate follows by assigning
one first derivative its supremum norm and the other its
$C^{0,\alpha}$ norm; their interpolation numerators are
$1-\gamma$ and $1+\alpha-\gamma$, respectively.

Freeze the coefficients in a chart of radius chosen using only
\eqref{eq:transmission-state-holder-all}.  Their oscillations in
supremum norm are then as small as desired, while their
$C^\gamma$ norms are bounded.  A principal coefficient error
times a top derivative has $C^{0,\alpha}$ norm bounded by a small
constant times $H$ plus a constant times $H^{\theta_2}$.
The same estimate after two tangential derivatives applies to the
value row in \eqref{eq:transmission-value-differentiation}; its
quadratic part is controlled by
\eqref{eq:transmission-product-interpolation}.  Apply
Lemma~\ref{lem:transmission-schauder} to cutoff fields with matched
cutoff values at a seam.  A second, slightly larger cutoff extends
coefficient differences without losing their small supremum norm.
Cutoff commutators contain only lower derivatives and have fixed
patch-dependent bounds.  Taking the maximum over a finite cover,
and using supremum interpolation for H\"older quotients with
separated points, gives
\begin{equation}
 H\leq \eta H+C\bigl(1+H^{\theta_1}+H^{\theta_2}
                         +H^{\theta_3}\bigr),\qquad
 0\leq\theta_3<1.
 \label{eq:transmission-global-absorption}
\end{equation}
Choose $\eta<1/2$.  This bounds $H$.  Constants for differentiated
cutoffs and source functions may depend on the finite parameter
choices, which is permitted here.

We must also justify using smooth-zero estimates for zeros initially
in \eqref{eq:transmission-domain-space}.  The following difference
quotient argument supplies that justification.  Off the seam,
ordinary scalar bootstrapping applies, estimating lapse and $w$
first and height next; at $S$ the $w$ conormal condition is its
ordinary homogeneous Neumann condition because the height gradient
is normal.  Near a seam flatten the face and take a tangential
difference quotient of the equations.  For the value relation use
the averaged first differential of $s$ between the two states.
For a fixed zero these coefficients are uniformly
$C^{2,\alpha'}$ and, on a sufficiently small patch and for
sufficiently small differences, are supremum-close to the same
frozen principal coefficients.  The difference quotients have
uniformly bounded one-lower norms from the starting regularity.
Apply the fixed-coefficient estimate to their cutoffs.  Principal
oscillations times top seminorms are absorbed; top supremum terms
are interpolated against the known one-lower norm.  All
commutators and differentiated explicit data involve only these
known orders.  This controls one tangential derivative in the same
column spaces.

The normal derivatives are recovered in a definite order.  Solve
the lapse and $w$ equations for their double normal derivatives;
their strictly positive normal coefficients permit division.
Then solve the height row for its double normal derivative, using
the newly controlled lapse derivatives.  Other second derivative
terms have at least one tangential index, which is already
controlled.  Iterating proves smoothness.  More explicitly, if
orders $m+3,\alpha'$ for height and $m+2,\alpha'$ for the other
fields are known, apply the same estimate to a tangential
difference quotient of $m$ tangential derivatives.  In the height
row, including its one residual derivative, every distributed
coefficient term other than the principal row on the increments
uses at most $m+2,\alpha'$ derivatives of $\mathscr U$.
The other interior rows start with at most one derivative of
$\mathscr U$ outside their own principal terms.  In the value row
the only new-order term is its first state differential on the
increment.  Thus the same argument gives the next derivatives with
the largest number of tangential indices.  Recover the others with
successively fewer tangential indices by the lapse and $w$
equations first, then the height equation.  These identities hold
on the open side and extend continuously to the face.  The quotient
near $S$ obeys \eqref{eq:transmission-quotient} at every step.

Consequently every zero of \eqref{eq:transmission-nonlinear-map}
is smooth, and \eqref{eq:transmission-global-absorption} bounds it
at the stronger exponent $\alpha$.  On a fixed truncation the
positive zero set over $0\leq u_*\leq1$ is compact in $X$:
the stronger H\"older bounds give precompactness, the residual map
is continuous, and the uniform positive floors for $N$ and $w$
keep limits in $X_+$.

\subsection{Fredholm index and continuation}

\begin{lemma}[Index zero for the linearized transmission problem]
\label{lem:transmission-fredholm}
The field derivative $D_X\mathscr F$ at each zero of
\eqref{eq:transmission-nonlinear-map} is Fredholm of index zero
between \eqref{eq:transmission-domain-space} and its specified
residual space.  Independently, the scalar density and adjoint
problems with smooth base metrics, fixed $C^{1,\alpha'}$ drift,
a fixed positive $C^{2,\alpha'}$ value ratio, natural flux matching,
inner Neumann and outer Dirichlet data are
Fredholm of index zero.  Their estimates may depend on derivatives
of the fixed ratio.
\end{lemma}

\begin{proof}
Discard compact lower-order terms in the linearization.  This is
legitimate with the column orders used above.  For example, a
variation of the trace target in the expanded lapse row contains
at most two derivatives of the height variation or one of the
other variations, and therefore maps compactly to its
$C^{0,\alpha'}$ residual.  Variations of $F/d_S$ lose one height
derivative by \eqref{eq:transmission-quotient}, which still leaves
this compactness.  In contrast, the first differential of
$N_o-s(N_i,\dd f)$ in the $C^{2,\alpha'}$ seam residual is
principal and is retained.  The principal height coupling to a
lapse derivative in its $C^{1,\alpha'}$ residual is also retained.

In this principal operator scale the background $\dd f$ continuously
to zero and recompute $q,\mathcal P,\beta,s,A$, the height coupling,
and the value-differential coefficients from that scaled background.
This scales coefficient fields, not the variations.  In particular
$ck=q-1$ in \eqref{eq:transmission-frozen-relations} holds all
along the path.  The states remain in a fixed positive ellipticity
range.  The local estimates of
Lemma~\ref{lem:transmission-schauder}, followed by localization,
give along this path
\begin{equation}
 \|v\|_X\leq C\bigl(\|L_\lambda v\|_{\mathscr Y}
                                  +\|v\|_0\bigr),
 \label{eq:transmission-linear-apriori}
\end{equation}
where $\|v\|_0$ is a sum of field supremum norms and $C$ is
uniform in the path parameter $\lambda$.

At zero background gradient the height decouples, and the lapse
and $w$ blocks have scalar value and positive conormal transmission.
Decrease each outer derivative coefficient in its flux relation to
zero, leaving the inner coefficient positive.  The determinant
$a+br$ in the proof of
Lemma~\ref{lem:transmission-schauder} stays positive, so
\eqref{eq:transmission-linear-apriori} remains valid.  At the
endpoint each scalar block is triangular: solve the inner Neumann
problem, then the outer Dirichlet problem with the trace prescribed
from the inner solution.  On each connected inner component the
scalar Neumann problem has equal kernel and compatibility-cokernel
dimensions, both one; the outer Dirichlet and height Dirichlet
problems have index zero.  Dividing by positive scalar principal
factors and discarding connection terms reduces this assertion to
ordinary Poisson theory.  Thus the endpoint index is zero.

We justify index propagation using only
\eqref{eq:transmission-linear-apriori}.  A sufficiently fine fixed
finite grid of field evaluations defines
$\mathfrak p:X\to\R^l$ with
$\|v\|_0\leq\eta\|v\|_X+C_\eta|\mathfrak p v|$.
Choose $\eta$ small in \eqref{eq:transmission-linear-apriori}.
The augmented operators
\begin{equation}
 \mathfrak A_\lambda=(L_\lambda,\mathfrak p):
                   X\longrightarrow\mathscr Y\times\R^l
 \quad\hbox{satisfy}\quad
 \|v\|_X\leq C\|\mathfrak A_\lambda v\|.
 \label{eq:transmission-augmented-bound}
\end{equation}
They are injective with closed range.  At an index-zero Fredholm
parameter their range has codimension $l$.  If parameters
$\lambda_j$ of this kind tend to $\lambda$, and the limit range
had codimension at least $l+1$, choose an $(l+1)$-dimensional
subspace $Z_0$ disjoint from that range.  At each $\lambda_j$
there is a unit vector $z_j\in Z_0\cap\operatorname{ran}
\mathfrak A_{\lambda_j}$.  Its preimages $v_j$ are bounded by
\eqref{eq:transmission-augmented-bound}.  A subsequence of $z_j$
converges in the finite-dimensional space to a unit vector $z$,
and operator-norm continuity gives
$\mathfrak A_\lambda v_j\to z$.  Closedness of the limit range
is a contradiction.  Its codimension is therefore finite.

The kernel of $L_\lambda$ is finite dimensional because
$\mathfrak p$ is injective on it.  Its range is closed and has
finite codimension: add the finite-dimensional factor
$\{0\}\times\R^l$ to the closed finite-codimensional augmented
range, and then quotient by that factor.  This sum is closed, and
the resulting quotient is exactly $\operatorname{ran}L_\lambda$.
Thus $L_\lambda$ is Fredholm also at the limit.  Openness and local
constancy of the Fredholm index, together with this closedness
argument starting from the endpoint, propagate index zero along
the whole path.  Restoring compact lower-order terms does not
change it.

For the independent scalar density problem take the fixed positive
$C^{2,\alpha'}$ ratio and fixed $C^{1,\alpha'}$ drift as coefficients.
Its local
principal boundary estimate is the scalar positive-ratio estimate
already proved, and its lower-order drift is compact at the scalar
column order two.  Localization uses only the stated coefficient
norms, so no approximation in a H\"older norm is needed.
The same outer-flux deformation and augmentation
argument applies without reference to a nonlinear zero or to an
a priori estimate for such zeros.  For its adjoint the value ratio
and positive derivative weight exchange their roles; the same
scalar combinations apply.  This proves both independent index
assertions.
\end{proof}

We now complete existence on a fixed truncation.  At $u_*=0$ the
height is identically zero.  Indeed its Dirichlet values are zero,
its trace target is zero, and testing its divergence equation by
$f$ gives $\int p\cdot\dd f=0$.  All scaled sources vanish and
$s_i=N_i$, so the lapse solves the scalar harmonic transmission
problem with ordinary value and normal-derivative matching, inner
flux $-b_*$ in the $n$ direction, and far value $1$.  The weak
form on the joined manifold is coercive after subtracting the
outer boundary value.  Equivalently its index is zero by
Lemma~\ref{lem:transmission-fredholm} and its homogeneous kernel
is zero by energy testing.  It therefore has a unique solution.
Comparison with $1$, using the outward inner flux $+b_*$, gives
$N\geq1$.  Scalar transmission regularity makes it piecewise
smooth.  The remaining equation is
$\divg(N^2\dd w)=0$ with matching value and flux, inner zero
flux, and far value $1$; hence $w=1$.

This initial zero is nondegenerate.  Its homogeneous linearization
is triangular: first
$\divg(N^2\dd\dot f)=0$ with zero height traces, then the
ordinary harmonic lapse transmission problem with homogeneous
boundary data, then the homogeneous $w$ divergence problem with
zero far value.  Energy testing makes each variation zero.
Terms differentiating the source cutoffs carry $u_*$ or vanish
because $\ell=0$; constitutive-coefficient variations in these
rows multiply $\dd f=0$ or $\dd w=0$.  Index zero now implies
invertibility of the full field derivative.

We supply the finite-dimensional continuation argument so that no
orientation or uniqueness assumption at the target parameter is
implicit.  Let $\mathcal K$ be the compact set of positive zeros
of \eqref{eq:transmission-nonlinear-map} over $[0,1]$.  By the
Fredholm assertion, a finite-dimensional subspace
$\mathfrak Z\subset\mathscr Y$ can be chosen so that
\begin{equation}
 D_X\mathscr F(X)+\mathfrak Z=\mathscr Y
 \label{eq:transmission-cokernel-span}
\end{equation}
at every point of $\mathcal K$: choose local cokernel complements
and use a finite cover.  Surjectivity persists on a neighborhood
$\mathcal O$ of $\mathcal K$.  If $Q$ is the quotient map to
$\mathscr Y/\mathfrak Z$, then
\begin{equation}
 \mathcal M=\{(u_*,v)\in\mathcal O:
                               Q\mathscr F(u_*,v)=0\}
 \label{eq:transmission-reduction-manifold}
\end{equation}
is a smooth finite-dimensional manifold of dimension
$\dim\mathfrak Z+1$.  The field derivative of $Q\mathscr F$
is onto, so the parameter projection on $\mathcal M$ is a
submersion.  Its only endpoint faces are therefore $u_*=0$ and
$u_*=1$.  Choose a relatively open neighborhood $\mathcal U$ of
$\mathcal K$ in $\mathcal M$ with compact closure contained in
$\mathcal O$ and with relative boundary disjoint from the zero
set.  This is possible in the finite-dimensional locally compact
manifold.  The norm of $\mathscr F$ has a positive minimum on
that relative boundary.

Apply Sard's theorem simultaneously to $\mathscr F|_{\mathcal M}$,
which takes values in $\mathfrak Z$, and its two endpoint faces.
For a sufficiently small common regular value
$z\in\mathfrak Z$, its preimage in $\mathcal U$ is a compact
one-manifold whose boundary lies only on the endpoint faces.
The implicit function theorem at the unique nondegenerate initial
zero gives exactly one point on the $u_*=0$ face.  Compactness of
$\overline{\mathcal U}$ excludes other points on that face for
small $z$: a sequence of such extra points would limit to an
additional zero, or to the initial zero in its uniqueness
neighborhood.  A compact one-manifold has an even number of
boundary points.  Consequently the $u_*=1$ face is nonempty.
Let a sequence of these regular values tend to zero and use
compactness of $\overline{\mathcal U}$ once more.  Its limiting
point is an exact positive zero at $u_*=1$.  This proves the
finite-truncation assertion of
Proposition~\ref{prop:transmission-existence}.

\subsection{Removal of the outer truncation}

Keep every internal parameter fixed and let $R\to\infty$ through
the solutions just obtained.  The lapse, slope, and low-order seam
bounds are locally uniform by
\eqref{eq:compact-bounds} and Lemma~\ref{lem:seam-holder}.  We
explain why the local upper bound for $w$ is also independent of
$R$, since the fixed-truncation energy bound alone does not imply
this assertion.

Choose a fixed sphere $\mathcal S$ outside all source supports.
On its exterior both $N_o$ and $x=N_ow$ are harmonic.  For $x$
this follows directly from the two equations:
\begin{equation}
 N_o\Delta_{g_o}(N_ow)=-P,
 \label{eq:transmission-exterior-x}
\end{equation}
so the conclusion holds wherever $P=0$, in particular beyond
$\mathcal S$.  Both fields have value $1$ at the truncating
boundary.  Positive radial superharmonic barriers on the AF end
of weight $|x'|^{-\beta}$, with any fixed $0<\beta<1$, bound
their deviations from $1$ by their bounded values at
$\mathcal S$ times such a weight.  Their construction uses
$\Delta_{g_o}|x'|^{-\beta}
=\beta(\beta-1)|x'|^{-\beta-2}
+O(|x'|^{-\beta-3})<0$ at large radius; enlarge $\mathcal S$
if necessary.  At a finite outer edge the zero deviation is
consistent with the positive barrier.

Suppose the supremum of $w$ on the fixed region inside
$\mathcal S$ were unbounded, and divide by this supremum $H_j$.
Exterior harmonic comparison for $x$, together with the global
positive floor for $N_o$, bounds the normalized $w$ everywhere
and gives a decay bound $C|x'|^{-\beta}$ for any subsequential
limit.  On compact sets its scalar forcing divided by $H_j$
tends to zero: $\Sigma_+x$ is bounded by its cutoff and the
finite state bounds, and $P$ has fixed finite strength.  The
coefficient matrices converge locally along a subsequence by the
H\"older estimates for $N$ and $\dd f$.  Scalar local
boundedness, H\"older compactness and weak energy estimates
then give a nonnegative limit $w_\infty$, with a nonzero
maximum on the fixed compact region, satisfying
\begin{equation}
 -\divg(A_\infty\dd w_\infty)=0,
 \qquad [w_\infty]_{\mathcal C}=0,
 \qquad [(A_\infty\dd w_\infty)_n]_{\mathcal C}=0,
 \qquad (A_\infty\dd w_\infty)_n|_S=0.
 \label{eq:transmission-normalized-w-limit}
\end{equation}
It tends to zero at infinity.  This is impossible: on arbitrarily
large compact joined domains, the weak maximum principle with
natural inner flux bounds it by the supremum on the distant
Dirichlet sphere, and that supremum tends to zero.  This argument
uses only the scalar joined divergence form and remains valid for
the limiting piecewise coefficients.  The contradiction proves
the required compact upper bound for $w$.

Without normalization the same exterior barriers show that every
limit has $N_o,x\to1$, and hence $w\to1$.  They also give
derivative estimates of every fixed order on annuli after scaling,
by ordinary scalar harmonic estimates for the fixed smooth AF
metric.  On the compact region containing the sources, the
H\"older estimates and the absorption argument above are uniform
in $R$: take an auxiliary fixed surrounding annulus, whose
derivatives are already bounded by these harmonic estimates, and
use the finite cover of compact patches inside it.  A diagonal
subsequence therefore converges in every fixed smooth norm on
each closed side on compact sets.  Its positive floors and its
boundary and transmission conditions persist.  This proves the
outer-limit assertion and completes
Proposition~\ref{prop:transmission-existence}.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/08-transmission-analysis.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/09-scalar-error.tex ====
\section{Scalar errors, smoothing, and free enclosures}
\label{sec:scalar-error}

Fix the compact data and the scalar-flat attachment of
Proposition~\ref{prop:af-attachment}.  All solutions in this section are
the actual solutions, with homotopy parameter equal to one, of
Proposition~\ref{prop:compact-solutions}.  In particular, the notation
\(N_i,N_o,f,s,w,x,W,T,\ell,\Sigma_\pm,P\) has the meaning given in
Section~\ref{sec:compact-construction}.  Constants can depend on the fixed
compact data, the positive numbers \(b_*,\delta_0\), the chosen exterior
collar, and the fixed value of \(L\).  They are independent of \(M\), of
the sufficiently small subsequent lapse cutoff, and of the sufficiently
large finite penalty strengths, unless stated otherwise.  None of the
arguments in this section restricts the topology of the compact part.

We first state the output needed in the final comparison.  The capped
manifold in this statement is auxiliary; no extension of the original
initial data across its boundary is asserted.

\begin{proposition}[Smooth AF comparison and two enclosures]
\label{prop:af-comparison}
Let \((\Omega_i,g,K)\) and \((\Omega_o,g_o)\) be the fixed inner data
and attachment of Proposition~\ref{prop:af-attachment}, with attachment
mass \(m_o\).  Given \(0<\delta<1\) and positive tolerances
\(\eta_m,\eta_v,\eta_g\), the parameters of
Proposition~\ref{prop:compact-solutions} can be chosen so that there is
a smooth complete connected one-ended AF manifold \((X,h')\), a fixed
topological cap set \(O\) with \(\partial O=S\), and precompact smooth
open sets \(E_0,E\) containing \(O\) in their interiors, with the
following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The boundaries of \(E_0\) and \(E\) are compact embedded minimal
surfaces.  Both sets are outward minimizing, their open exteriors are
connected, and
\begin{equation}
 \Per_{h'}(E_0)\le \Per_{h'}(E).
 \label{eq:unsafe-two-perimeters}
\end{equation}
The set \(E_0\) is a largest perimeter minimizing enclosure of \(O\)
alone, is strictly outward minimizing, and every component of \(E_0\)
meets \(O\).
\item On the original side of the caps,
\begin{equation}
 \Scal_{h'}\ge-2,\qquad
 \Scal_{h'}\ge0\quad\hbox{on }X\setminus E,\qquad
 m_{h'}\le m_o+\eta_m .
 \label{eq:unsafe-comparison-output}
\end{equation}
In particular the closed exterior of \(E\) is smooth and complete,
has exactly one AF end, and has outer minimizing minimal boundary.
For any fixed \(1/2<\beta<1\), its end has metric decay
\(O_2(|x'|^{-\beta})\) and scalar decay
\(O(|x'|^{-2-2\beta})\).
\item Identify the two original base pieces with their images in
\(X\setminus\operatorname{int}O\), smoothly on each closed side of
the seam, and let \(g_{\rm base}\) denote their respective metrics
\(g,g_o\).  For the piecewise graph metric
\(h=x\bar g\), where \(\bar g=g+s^2\dd f^2\) inside and
\(\bar g=g_o\) outside, one has
\begin{equation}
 (1-\eta_g)h\le h'\le(1+\eta_g)h,\qquad
 \bar g\ge g_{\rm base},\qquad
 \dd V_{\bar g}=W\,\dd V_{\rm base},
 \label{eq:unsafe-metric-output}
\end{equation}
after decreasing \(\eta_g\) below one if necessary, and
\begin{equation}
 \int_{\{x<1-\delta\}}W\,\dd V_{\rm base}<\eta_v.
 \label{eq:unsafe-bad-volume-output}
\end{equation}
The scalar lower bound and the topology of \(X\setminus
\operatorname{int}O\) are independent of the large parameters.
\end{enumerate}
The order of choice is the following: choose \(b_*,\delta_0\) for the
mass tolerance, then \(\delta\), the thin exterior omitted collar and
\(L\) for the volume tolerance, then \(M\), then the lapse cutoff and
the finite penalty strengths.  The corner smoothing is chosen last,
separately for each resulting fixed solution.  In the enclosure
construction a large fixed scaled ball radius is chosen before \(M\).
\end{proposition}

We prove the proposition in several steps.  Throughout, a closed unsafe
set is used, so excluding it from the new exterior also excludes limits
of the regions in which the scalar sign is uncertain.

\subsection{Uniform geometry at unsafe points}

Define \(\mathcal U_M\subset\overline\Omega_i\) by the closed
versions of the four trace-cutoff inequalities in
\eqref{eq:compact-trace-cutoff}, together with \(N_i\ge\delta_0\).
The quotient at \(S\) has the continuous interpretation specified
there.  This is a compact set for each fixed solution.  The scalar
identity \eqref{eq:compact-scalar} and the definition of \(\Sigma_-\)
show that
\begin{equation}
 \Scal_h\ge0\quad\hbox{off }\mathcal U_M
 \quad\hbox{on each smooth side of the seam}.
 \label{eq:unsafe-scalar-support}
\end{equation}
Indeed the only possibly negative term is
\(-\ell^2(1-\zeta(N_i))\); it vanishes when the trace cutoff is
inactive or when \(N_i\le\delta_0\).

At \(P\in\mathcal U_M\), set
\begin{equation}
 N_0=N_i(P),\qquad r_P=x(P)^{-1/2}.
 \label{eq:unsafe-scale}
\end{equation}
Since \(f(P)\ge9M/10\), \(z(P)\le1/2\), and
\(w(P)\ge M^{9/10}\),
\begin{equation}
 x(P)\ge\frac{\sqrt3}{2}\delta_0 M^{9/10},\qquad
 r_P\le C M^{-9/20}.
 \label{eq:unsafe-radius-bound}
\end{equation}
The negative term in the scalar identity is bounded independently of
\(M\), so \eqref{eq:unsafe-scalar-support} also gives
\(\Scal_h\ge-1\) everywhere on the two smooth pieces for all
sufficiently large \(M\).

\begin{lemma}[Geometry at the unsafe scale]
\label{lem:unsafe-blowup}
There is a constant \(C\ge1\) with the following property.  For every
fixed \(R<\infty\), all sufficiently large \(M\), and every
\(P\in\mathcal U_M\), the base ball of radius \(Rr_P\) about \(P\)
does not meet the seam.  In interior charts, or in charts flattened at
\(S\) when necessary, whose unit of base length is \(r_P\),
\begin{equation}
 C^{-1}\Id\le h\le C\Id
 \label{eq:unsafe-uniform-metric}
\end{equation}
on the corresponding ball, restricted to the original side of \(S\).
The same comparison holds after reflection across \(S\).  The constant
\(C\) is independent of \(R\); the required lower bound for \(M\)
may depend on \(R\).  Equivalently, the order of the quantifiers is
\begin{equation}
 \exists C\ \forall R<\infty\ \exists M_0(R)\
 \forall M\ge M_0(R)\ \forall P\in\mathcal U_M.
 \label{eq:unsafe-quantifiers}
\end{equation}
\end{lemma}

\begin{proof}
We first control the normalized lapse without a slope bound.
Height compactness and a sublevel argument then exclude positive
limiting heights and a seam at bounded scaled distance.
Only after this do we use the gradient estimate to obtain the flat
graph limit. Finally a truncated Harnack argument controls the
conformal factor with a constant independent of the fixed scaled radius.

It is enough to examine an arbitrary sequence \(M\to\infty\) and
marked centers \(P\in\mathcal U_M\).  Write \(r=r_P\).  Use smooth
base charts, with Fermi charts at a boundary, and rescale their lengths
by \(r\); equivalently use the base metric \(g_r=r^{-2}g\).  On each
fixed scaled compact set the base metrics converge smoothly to the
Euclidean metric, or to the Euclidean metric on a half-space.  The two
compact boundary faces \(S\) and \(\mathcal C\) have positive base
separation, so at most one can remain at a bounded scaled distance.

\smallskip\noindent\emph{Lapse bounds before slope bounds.}
We first obtain lapse estimates without using height-gradient bounds.
Testing the height equation on nested base balls with a cutoff and a
height function bounded in absolute value by \(M\) gives, in scaled
volume units,
\begin{equation}
 \int_{B_H}(W^2-1)\,\dd V_{g_r}
 \le C_H\frac{M}{r}
       \int_{B_{H'}}N_i\,\dd V_{g_r},\qquad H<H'.
 \label{eq:unsafe-local-height-energy}
\end{equation}
For a ball reaching a Dirichlet face, subtract its constant boundary
height before testing; the boundary term then vanishes.  To verify the
estimate, use \(p\cdot\dd f=W^2-1\), \(|p|\le N_i\),
\(|T|\le C\), and a cutoff derivative of size \(C_H/r\).
These are also the only estimates required if the ball meets a face.

Put \(U=N_i/N_0\), and let \(A_{H'}\) be a supremum of this
normalized lapse on a larger fixed scaled patch.  The scalar forcing
in its rescaled equation obeys
\begin{align}
 \left\|\frac{r^2}{N_0}(\Sigma_++\Sigma_-)\right\|_{L^2(B_H,g_r)}
 &\le C_H\left(r^2+
       M^{1/2}r^{3/2}N_0^{-1/2}A_{H'}^{1/2}\right) \\
 &\le C_H\left(r^2+M^{-7/40}A_{H'}^{1/2}\right).
 \label{eq:unsafe-scaled-source}
\end{align}
Here \(\Sigma_+\le LW\), \(|\Sigma_-|\le C\delta_0\),
\(N_0\ge\delta_0\), and \eqref{eq:unsafe-local-height-energy} were
used.  The divergence drift has size \(O(r)\).  The natural flux
error at \(S\) has size \(O(rb_*/N_0)\); a bounded extension of the
normal datum incorporates it in the divergence forcing.

If no seam is present on the patch, the inhomogeneous Harnack estimate
for \(U\), including natural-flux reflection at \(S\), bounds the
smaller supremum by a constant times its infimum plus the right side
of \eqref{eq:unsafe-scaled-source}.  Choose the smaller patch to
contain the center, or use a boundary-foot ball containing it, so that
the infimum is at most \(U(P)=1\).  Thus, on finitely many nested
patches,
\begin{equation}
 A_H\le C_H\left(1+r^2+M^{-7/40}A_{H'}^{1/2}\right).
 \label{eq:unsafe-nonseam-iteration}
\end{equation}
The crude bound \(N_i\le C(L)M\) gives \(A_{H'}=O(M)\).
The exponent of \(M\) improves by
\(a\mapsto\max(0,a/2-7/40)\).  Two applications give
\(1\mapsto13/40\mapsto0\).  We have therefore obtained local
boundedness by a finite iteration, before asserting that the forcing
tends to zero.

There is a slightly different normalization argument if the seam has
bounded scaled distance.  Reflect the exterior lapse into the inner
side in the two base Fermi collars, and set
\(V=U+N_o^{\rm refl}/N_0\).  Use the inner density-cometric matrix
for the summed equation.  The common induced metric implies that the
two matrices differ by \(O(r|t|)\) at scaled distance \(|t|\) from
the face.  Positivity and the interior gradient estimate for the
harmonic exterior lapse on balls of radius comparable to \(|t|\)
give
\begin{equation}
 |(a_i-a_o^{\rm refl})\nabla(N_o^{\rm refl}/N_0)|
 \le Cr(N_o^{\rm refl}/N_0).
 \label{eq:unsafe-seam-drift}
\end{equation}
The exterior is harmonic on the fixed omitted collar, which contains
every such scaled patch for large \(M\).  The inner drift obeys the
same bound with \(U\).  Consequently the total correction is a
bounded divergence drift of size \(Cr\) times \(V\), with zero
natural flux at the face, and the scalar forcing is still bounded by
\eqref{eq:unsafe-scaled-source} with a larger supremum of \(V\).

The infimum in Harnack for \(V\) must be anchored to the inner
center value.  Here is that step explicitly.  Let \(m\) be the
infimum of \(V\) on a fixed smaller half-patch containing the
center and a central disk in the face.  On the face,
\(N_o=dN_i\), \(0<d\le1\), whence \(U\ge m/2\).
Choose a fixed smooth half-domain with that flat disk in its boundary
and containing the center.  The harmonic measure of a smaller disk
at the center is bounded below by a fixed positive number; this
remains true as the center approaches that disk.  Write \(U=H+e\),
where \(H\) is the harmonic replacement with the same boundary
values.  The zero-Dirichlet estimate for \(e\), using divergence
forcing in \(L^p\), \(3<p<6\), and scalar forcing in \(L^2\),
gives
\begin{equation}
 \|e\|_\infty\le
 C_H\left(rA_{H'}+r^2+M^{-7/40}A_{H'}^{1/2}\right).
 \label{eq:unsafe-replacement-error}
\end{equation}
The remaining boundary values of \(H\) are nonnegative, and
\(U(P)=1\).  Harmonic measure therefore bounds \(m\) by a
constant times one plus this error.  Combining this with the Harnack
estimate for the reflected sum yields
\begin{equation}
 A_H\le C_H\left(1+rA_{H'}+r^2+
                 M^{-7/40}A_{H'}^{1/2}\right).
 \label{eq:unsafe-seam-iteration}
\end{equation}
Starting again from \(O(M)\), the exponent improves by
\(a\mapsto\max(0,a-9/20,a/2-7/40)\).  Three nested applications
give \(1\mapsto11/20\mapsto1/10\mapsto0\).  Thus the sum is
locally bounded.  This argument uses no derivative of the seam trace
ratio and no separate trace compactness for either lapse.

The forcing in \eqref{eq:unsafe-scaled-source} now tends to zero.
Local Laplace estimates give \(W^{1,p}\) bounds, \(p>3\), and
H\"older compactness, including reflected natural boundary estimates
when appropriate.  Without a seam, every subsequential limit is a
positive harmonic function on all of \(\R^3\), or on a half-space
with zero Neumann data.  Positive harmonic Liouville, after reflection
in the latter case, and the center normalization show that the limit
is one.  At a seam the reflected sum has instead a positive constant
limit.  Its constant is positive because it is at least the inner
center value.  Its trace lower bound, the relation \(U\ge V/2\)
on the face, and harmonic replacement with the error
\eqref{eq:unsafe-replacement-error} tending to zero give a positive
lower bound for the inner lapse up to the face.  On compact open
inner sets it converges, in a smaller H\"older exponent, to a smooth
positive harmonic function.  These conclusions hold on every fixed
patch by diagonal extraction.  In particular both upper and positive
lower lapse bounds are now available in every case.

\smallskip\noindent\emph{Height compactness.}
We turn to the height, setting
\begin{equation}
 \mathcal F=\frac{N_0(M-f)}{r},\qquad
 \mathcal U=\frac{N_i}{N_0},\qquad \widetilde s=\frac{s}{N_0}.
 \label{eq:unsafe-normalized-height}
\end{equation}
It is nonnegative and \(\mathcal F(P)\le M^{-1}\) by the closed
cutoff inequality.  Its flux in the metric \(g_r\) is
\begin{equation}
 \mathfrak p=\widetilde s^2\nabla_{g_r}\mathcal F
       =-\frac r{N_0}p,\qquad
 \divg_{g_r}\mathfrak p=-rT\mathcal U=o(1).
 \label{eq:unsafe-height-equation}
\end{equation}
The constitutive rule is unchanged:
\(\widetilde s^2+|\mathfrak p|_{g_r}^2=\mathcal U^2\).
If \(\mathcal U\) is frozen at its actual value, the flux is
strictly monotone in its gradient argument \(\xi\), and the positive
upper and lower lapse bounds give
\begin{equation}
 \langle\mathfrak p(\xi),\xi\rangle
   \ge c\min(|\xi|^2,|\xi|),\qquad
 |\mathfrak p(\xi)|^2
   \le C\langle\mathfrak p(\xi),\xi\rangle.
 \label{eq:unsafe-flux-coercivity}
\end{equation}
For example, in an orthonormal frame
\(\mathfrak p(\xi)=a\xi\), where
\(a+a^2|\xi|^2=\mathcal U^2\); these assertions follow directly
by differentiation and by considering \(|\xi|\le1\) and
\(|\xi|\ge1\).

We require a compactness argument for possibly very steep graphs.
In the product of a scaled base chart with a height line, the bounded
vector field \((-\mathfrak p,1)\) has bounded divergence and its
pairing with the upward unit graph normal is bounded below by a
positive constant, by \eqref{eq:unsafe-flux-coercivity}.  Gauss--Green
applied to the subgraph and supergraph in product balls gives, for
either side's volume function \(V(a)\), an upper bound
\(C(V'(a)+V(a))\) for the graph perimeter in that ball.  It also
gives local perimeter bounds by using fixed surrounding balls.
The four-dimensional isoperimetric inequality, at sufficiently small
radii to absorb the volume term, yields
\(V'(a)\ge cV(a)^{3/4}\).  Thus at every interior graph point both
sides have volume at least \(ca^4\).  These constants are uniform
on each of the fixed patches under consideration.

At \(S\), \(\mathcal F=0\).  For product balls at heights bounded
away above zero, extend the subgraph as the empty set across the
face.  There is no new face perimeter, because its trace is empty
there.  The same subgraph density estimate is therefore valid even
when the ball reaches \(S\).  BV compactness on open inner product
patches and vertical nesting now give a subsequence such that all
bounded height truncations converge in local measure to those of an
extended measurable function \(\mathcal F_\infty\ge0\), whose
value is allowed initially to be \(+\infty\).

We next prove that a nonzero limiting height forces a positive lower
bound.  On a ball \(B_H\) contained in the inner side, test
\eqref{eq:unsafe-height-equation} with
\(\chi^2(k-\mathcal F)^+\), where \(\chi=1\) on \(B_h\).
By \eqref{eq:unsafe-flux-coercivity} and Young's inequality,
\begin{equation}
 \int_{B_h}\min(|\dd(k-\mathcal F)^+|^2,
                        |\dd(k-\mathcal F)^+|)
 \le C\left(\frac{k^2}{(H-h)^2}+\varepsilon_M k\right)
              |\{\mathcal F<k\}\cap B_H|,
 \label{eq:unsafe-sublevel-energy}
\end{equation}
where \(\varepsilon_M\to0\) on the fixed patch.  The same test is
allowed on a half-patch with extension by zero across a Dirichlet
face whose height exceeds \(k\).  On the portion with derivative at
most one, use Cauchy--Schwarz; on its complement use the linear part
of the energy.  The \(W^{1,1}\) Sobolev inequality for an
intermediate cutoff then gives, for \(0<l<k\),
\begin{equation}
 (k-l)|\{\mathcal F<l\}\cap B_h|^{2/3}
 \le C\left(\frac{k}{H-h}+
             \frac{k^2}{(H-h)^2}+o(1)\right)
                |\{\mathcal F<k\}\cap B_H|.
 \label{eq:unsafe-sublevel-sobolev}
\end{equation}
The error here includes \((\varepsilon_M k)^{1/2}\) and
\(\varepsilon_M k\), which tend to zero for fixed levels.

For completeness, take radii decreasing from \(H\) to \(H/2\)
and levels decreasing from \(k\) to \(k/2\) geometrically.  First
take \(M\) sufficiently large at the fixed \(H,k\) to absorb the
forcing into the first cutoff term.  This also absorbs it at every
later step, since the level remains at least \(k/2\) and the cutoff
derivatives only increase.  After dividing volumes by \(H^3\),
\eqref{eq:unsafe-sublevel-sobolev} has the form
\begin{equation}
 v_{j+1}^{2/3}\le C^{j+1}(1+k/H)v_j.
 \label{eq:unsafe-sublevel-iteration}
\end{equation}
For \(0<k\le H\), a sufficiently small initial fraction, with a
threshold independent of \(k/H\), therefore forces \(v_j\to0\).
If \(\mathcal F_\infty\) is not zero almost everywhere, choose
a density ball of one of its positive superlevel sets and then a
small fixed \(k>0\).  Convergence of truncations supplies that
small initial fraction.  Consequently \(\mathcal F\ge k/2\)
on a smaller open inner ball for all sufficiently large indices.

This lower bound propagates to compact subsets of the connected
limiting inner domain.  To justify propagation for the capped flux,
choose a fixed smooth connecting subdomain with a small nonnegative
Dirichlet bump on a boundary portion inside the known positive ball
and zero data elsewhere.  On this subdomain the actual lapse fields
converge in a H\"older exponent to smooth positive coefficients.
The flux is smooth and uniformly elliptic near zero gradient.  The
Dirichlet inverse of its linearization at zero maps divergence
\(C^\gamma\) data to \(C^{1,\gamma}\); contraction in a small
\(C^{1,\gamma}\) ball therefore supplies small-gradient barriers,
uniformly for the sequence, with divergence a positive constant
tending to zero that dominates the right side of
\eqref{eq:unsafe-height-equation}.  Their limiting solutions have
positive interior values by the strict maximum principle.  The
constructed boundary values are below those of \(\mathcal F\).
Strict gradient-flux monotonicity, testing the positive part of the
barrier minus \(\mathcal F\), proves comparison in the required
direction.  A finite chain of such domains and a finite covering
prove propagation on every compact inner subset.

In the \(S\) alternative, the lapse convergence is also H\"older
up to that face.  The connecting subdomains may then reach a flat
boundary patch, with zero data on that portion.  The limiting
barrier is smooth there and has strictly positive inward derivative
by Hopf's lemma.  The \(C^1\) convergence of the barriers gives
a uniform lower bound by a positive constant times inward scaled
distance.  This contradicts the closed unsafe condition
\begin{equation}
 \frac{\mathcal F(P)}{d_S(P)/r}\le M^{-1},
 \label{eq:unsafe-boundary-touchdown}
\end{equation}
with the normal derivative interpretation if \(P\in S\).
An interior limiting center is already excluded by
\(\mathcal F(P)\le M^{-1}\).  Thus a nonzero limiting height
is impossible in these two alternatives.

\smallskip\noindent\emph{Excluding seam limits.}
If the center approaches the seam, the boundary height is instead
\(N_0M/r\to\infty\).  A positive limiting height propagates
through the open half-space as above.  On a fixed ball reaching the
face, omit first a thin strip adjacent to the face.  The lower
bound on the remaining compact set, followed by a small choice of
\(k\), makes the starting sublevel fraction arbitrarily small.
Apply \eqref{eq:unsafe-sublevel-iteration} with the negative part
extended by zero across the seam.  It supplies a positive lower
bound up to the central face patch and contradicts center
touchdown.  This step does not require H\"older convergence of
the individual lapse traces.

The zero almost everywhere alternative is also impossible at a
seam.  On a fixed half-patch choose a smooth nonnegative function
\(b\) with amplitude \(k\), derivative \(O(k)\), positive by
\(ck\) on a central face disk and zero near the other edges.
The test \((b-\mathcal F)^+\) has zero boundary trace, since the
seam height tends to infinity.  Monotonicity and
\eqref{eq:unsafe-flux-coercivity} give
\begin{equation}
 \int_{\{\mathcal F<b\}}\min(|\dd\mathcal F|^2,
                                      |\dd\mathcal F|)
       \le Ck^2+o(1)k.
 \label{eq:unsafe-seam-energy}
\end{equation}
The function \(v=\min(\mathcal F,b)\) has trace \(b\) on
that disk and tends to zero in measure in the interior.  On almost
every line normal to a smaller central disk, the fundamental
theorem of calculus, followed by averaging over interior slices of
a fixed thin collar \(D_\varepsilon\), gives
\(\int_{D_\varepsilon}|\dd v|\ge ck-o(1)\).  Here convergence
in measure suffices because \(0\le v\le k\).  Since
\(\min(t^2,t)\ge t-1/4\) for \(t\ge0\), the left side of
\eqref{eq:unsafe-seam-energy} is at least
\begin{equation}
 ck-Ck|D_\varepsilon|-\tfrac14|D_\varepsilon|-o(1).
 \label{eq:unsafe-seam-trace-cost}
\end{equation}
Choose first \(k>0\) small enough that \(ck\) dominates
\(Ck^2\), and then the collar thin enough.  This contradicts
\eqref{eq:unsafe-seam-energy}.  All possible seam limits have now
been excluded.  Consequently
\begin{equation}
 \dist_g(P,\mathcal C)/r_P\longrightarrow\infty,
 \qquad \mathcal F_\infty=0
 \quad\hbox{almost everywhere in the remaining limits}.
 \label{eq:unsafe-seam-escape}
\end{equation}

\smallskip\noindent\emph{Uniform height and slope convergence.}
The height convergence is in fact uniform on every scaled compact
set, including up to \(S\).  Otherwise continuity and local
convergence in measure produce graph points at some fixed positive
height in a slightly larger patch.  A fixed small product ball at
such a point has uniformly positive subgraph volume by the density
bound proved above.  But that volume tends to zero because the
limiting height is zero.  If the points approach \(S\), use the
empty extension across the face; alternatively omit a thin boundary
strip before taking the limit.  The strip has too little product
volume to account for the density lower bound.  This excludes
positive spikes as well as bounded positive plateaus.

We can now apply the differential gradient estimate
Lemma~\ref{lem:gradient-maximum}.  Under the rescaling above the
synthetic endomorphism and trace target become \(-rK\) and
\(-rT\), and the lapse current becomes \((r^2/N_0)B\), so the
same identity applies.  Maximize
\begin{equation}
 -\log\widetilde s+j(\mathcal F)+k_0\log\chi,
 \label{eq:unsafe-local-gradient-test}
\end{equation}
where \(k_0\ge2\), \(\chi\) is a fixed local cutoff, and
\(j',j''\) are positive, bounded, and bounded away from zero on
the now bounded height range.  If \(m=|\dd\mathcal F|_{g_r}\),
the constitutive relations give
\begin{equation}
 qm^2=\frac{J}{\widetilde s^2}
       =\frac{JW^2}{\mathcal U^2},\qquad
 (1-y)^2(j'm)^2=\frac{(j')^2z^2}{\mathcal U^2}.
 \label{eq:unsafe-scaled-gradient-algebra}
\end{equation}
At an interior maximum with \(y\ge1/2\), the positive leading
term therefore controls \(W^2\).  The source term has size
\(r^2\Sigma_+/N_i\le CW\), and localization costs
\(C\chi^{-2}\).  Hence
\begin{equation}
 cW^2\le C(1+W+\chi^{-2}).
 \label{eq:unsafe-slope-bound}
\end{equation}
At \(S\), \eqref{eq:calculus-boundary-lapse}, \(B_n=-b_*\),
\(k_T\hat z\ge0\), and \(H_g\le0\) give
\(\partial_n(-\log\widetilde s)\ge0\) for the inward normal.
Also \(\partial_n\mathcal F=m\ge0\).  The derivative condition
at a boundary maximum in \eqref{eq:unsafe-local-gradient-test}
therefore gives \(j'm\le C/\chi\), which controls that case.
When \(y<1/2\), \(W\le\sqrt2\) directly.  The maximum test
and \(k_0\ge2\) thus bound \(W\) on a smaller patch.

After this finite slope bound, differentiating the height divergence
equation gives uniformly elliptic scalar divergence equations for
its first derivatives.  Their divergence forcing is controlled in
\(L^p\), \(p>3\), by the normalized lapse estimates.  Odd
reflection of \(\mathcal F\), with even reflection of lapse and
the Fermi metric, gives the same estimate at \(S\).  The resulting
gradient H\"older estimates and the uniform height convergence
imply
\begin{equation}
 \mathcal U\longrightarrow1,\qquad
 \dd\mathcal F\longrightarrow0,\qquad
 \widetilde s\longrightarrow1
 \label{eq:unsafe-flat-graph-limit}
\end{equation}
on every fixed scaled compact set.

\smallskip\noindent\emph{The conformal factor.}
It remains to control the conformal factor uniformly in the chosen
fixed radius.  Put \(w'=w/w(P)\).  On every fixed scaled patch,
the height just proved gives \(f\ge9M/10\), hence \(w\ge T_1\)
with \(T_1=M^{9/10}\).  The lapse is eventually at least
\(\delta_0/2\); choose the subsequent cutoff
\(\eps<\delta_0/8\).  Also
\(x=sw\to\infty\) there.  The volume source and the low-lapse
penalty are consequently absent.  The positive function \(w'\)
is a supersolution for its uniformly elliptic divergence principal
part, while
\(v'=(w'-a)^+\), \(a=4T_1/w(P)\), is a subsolution: above
that truncation the remaining height penalty vanishes as well.
Weak Harnack for \(w'\), combined with the local supremum bound
for \(v'\) with the same positive mean exponent, gives on balls
with fixed nesting ratios
\begin{equation}
 \sup_{B_R}w'
 \le a+C\left(\frac1{|B_{2R}|}\int_{B_{2R}}(w')^p\right)^{1/p}
 \le a+C'\inf_{B_R}w'\le(4+C')\inf_{B_R}w'.
 \label{eq:unsafe-w-harnack}
\end{equation}
The last step uses \(w'\ge T_1/w(P)=a/4\).  Natural zero flux
permits reflection at \(S\).

These constants can be chosen independently of the fixed radius.
Indeed, on each arbitrarily large fixed patch the normalized
principal matrices tend to the identity by
\eqref{eq:unsafe-flat-graph-limit}.  If the boundary is far enough,
use concentric balls about the center with universal radius ratios.
Otherwise use comparable larger balls about its boundary foot in a
single flattened chart, and reflect.  The smaller ball contains both
the center and the requested region.  Uniform ellipticity with fixed
constants and dilation invariance of the homogeneous estimates give
one common constant in \eqref{eq:unsafe-w-harnack}.  The index from
which that constant applies may depend on the radius.  Since
\(w'(P)=1\), it follows that \(C^{-1}\le w'\le C\).

In the rescaled coordinates the metric is
\begin{equation}
 h=\frac{x}{x(P)}
       \left(g_r+\widetilde s^2\dd\mathcal F^2\right),
 \qquad
 \frac{x}{x(P)}=\frac{\widetilde s}{\widetilde s(P)}w'.
 \label{eq:unsafe-rescaled-metric}
\end{equation}
Here, as usual, the expression denotes its pullback by the coordinate
rescaling.  Equations~\eqref{eq:unsafe-flat-graph-limit} and
\eqref{eq:unsafe-w-harnack} prove
\eqref{eq:unsafe-uniform-metric}.  At \(S\) the mixed metric
entries vanish, so reflection preserves this comparison.  If the
uniform statement of the lemma failed, a sequence violating it would
admit precisely the subsequences just analyzed, a contradiction.
\end{proof}

\subsection{Mass and a fixed enclosing comparator}

On the AF end outside a common compact support, \(N_o\) and
\(x=N_ow\) are harmonic and tend to one.  Radial barriers of
weight \(|x'|^{-\beta}\), \(1/2<\beta<1\), and estimates on
annuli rescaled to unit size give
\begin{equation}
 N_o=1+O_2(|x'|^{-\beta}),\qquad
 x=1+O_2(|x'|^{-\beta}).
 \label{eq:unsafe-af-decay}
\end{equation}
The constants in these end bounds can depend on the fixed solution.
Thus \(h=xg_o\) is AF of this order.  Since \(g_o\) is scalar
flat and \(x\) is harmonic there, its scalar curvature is
\(O(|x'|^{-2-2\beta})\), an integrable decay rate.

The ADM flux of the conformal metric error gives
\begin{equation}
 m_h-m_o=-\frac1{8\pi}
           \lim_{R\to\infty}\int_{|x'|=R}\partial_nx\,\dd A_{g_o}.
 \label{eq:unsafe-adm-flux}
\end{equation}
One may replace the conormal-area product by its Euclidean version:
the error is quadratic in the decaying fields and tends to zero.
On this end the current in \eqref{eq:calculus-compact-current} is
\(Q=N_o\nabla x+(1-x)\nabla N_o\), whose flux has the same
limit as \(\nabla x\).  Integration of its divergence over both
pieces, cancellation of the matched seam fluxes with their common
base area form, and the outward flux \(+b_*\) at \(S\) yield
\begin{equation}
 8\pi(m_h-m_o)
   =\int_S b_*\,\dd A_g
      +\int\big((x-1)\Sigma_++P-\Sigma_-\big)\,\dd V_{\rm base}.
 \label{eq:unsafe-mass-budget}
\end{equation}
This also proves existence of the ADM limit: the current divergence
has compact support.  By Lemma~\ref{lem:compact-penalties},
\begin{equation}
 \limsup_{M\to\infty}(m_h-m_o)\le C(b_*+\delta_0).
 \label{eq:unsafe-mass-limsup}
\end{equation}
The constant on the right uses only the fixed compact data.  The
penalty cutoff makes \((x-1)\Sigma_+\le0\), and the positive
penalty costs tend to zero after the subsequent choices specified
in Section~\ref{sec:compact-construction}.

We will need a competitor whose area remains bounded as \(M\)
increases.  Fix a sphere \(\mathcal S_1\) in the attached end
beyond all source supports.  Let \(\psi_1\) be its exterior
capacitary potential: it is harmonic, is one on \(\mathcal S_1\),
and tends to zero at infinity.  Existence follows by solving on
compact annuli, using the preceding barriers, and taking a limit.
The function \(\gamma_1=-\partial_n\psi_1\) is strictly positive
on the sphere by Hopf's lemma.  Green's formula for \(x-1\) and
\(\psi_1\) gives
\begin{equation}
 \int_{\mathcal S_1}\gamma_1(x-1)\,\dd A_{g_o}
       =-\lim_{R\to\infty}\int_{|x'|=R}\partial_nx\,\dd A_{g_o}
       =8\pi(m_h-m_o).
 \label{eq:unsafe-capacity-comparator}
\end{equation}
The boundary term at infinity in the first Green identity vanishes
by \(\beta>1/2\); alternatively apply the harmonic flux identity
for \(x\) on the intervening annulus.  Positivity of \(x\), the
positive minimum of \(\gamma_1\), and the upper mass bound imply
\begin{equation}
 \Area_h(\mathcal S_1)
       =\int_{\mathcal S_1}x\,\dd A_{g_o}\le C_*.
 \label{eq:unsafe-comparator-bound}
\end{equation}
For the fixed earlier choices, \(C_*\) is independent of large
\(M\) and of the later parameters.  It is also independent of
the scaled radius of the obstacles introduced below.

\subsection{Removing the corner}

The collar mollification and conformal correction below use Miao's
corner-smoothing method \cite{Miao2002}. We give the local argument for
the present unsafe-set and cap construction, rather than invoke his
global positive mass theorem.

\begin{lemma}[Corner smoothing with controlled mass]
\label{lem:corner-smoothing}
For each sufficiently large fixed solution, the joined metric \(h\)
admits smooth AF replacements \(h'\) on the original side of the
caps such that, with errors made arbitrarily small separately at
that solution,
\begin{equation}
 h'=(1+o(1))h\ \hbox{as quadratic forms},\qquad
 m_{h'}=m_h+o(1),\qquad H_{h'}(S)<0.
 \label{eq:unsafe-smoothing-estimates}
\end{equation}
They satisfy \(\Scal_{h'}\ge0\) off \(\mathcal U_M\) and
\(\Scal_{h'}\ge-2\) everywhere on the original side of the
caps.  Their AF metric and scalar decay are those of
\eqref{eq:unsafe-af-decay}.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:unsafe-blowup} the closed unsafe set is disjoint
from the seam for large \(M\).  Glue the two metrics using their
own Gaussian normal collars.  In the resulting smooth manifold
structure the joined metric is continuous and has the form
\(\dd t^2+\lambda(t)\), piecewise smooth, with the inner side
at \(t<0\).  The common induced metric and the mean-curvature
inequality in \eqref{eq:compact-mean-curvature} give
\begin{equation}
 \frac12\tr_{\lambda(0)}
              \big(\lambda'(0^+)-\lambda'(0^-)\big)\le0.
 \label{eq:unsafe-corner-sign}
\end{equation}
This change of collar identification retains a smooth identification
of each original closed piece with its image; comparisons with the
base metric will always use those identifications.

Mollify \(\lambda\) in \(t\) with a nonnegative smooth even
kernel of radius \(a\), and interpolate back to the original
smooth metrics in \(2a\le|t|\le3a\).  The resulting \(h_a\)
converges uniformly to \(h\); its first normal derivatives and
fixed tangential derivatives are bounded.  Its second normal
derivative is the jump of \(\lambda'\) times the mollifier plus
a uniformly bounded remainder.  The interpolation has the same
bound, since on those smooth one-sided regions the mollification
error is \(O(a^2)\) before taking the two cutoff derivatives.
In the hypersurface scalar-curvature formula the only potentially
unbounded term is \(-\tr_\lambda\lambda''\).  Replacing the
inverse metric in its jump term by \(\lambda(0)^{-1}\) costs
\(O(a)O(a^{-1})=O(1)\); the leading term has favorable sign by
\eqref{eq:unsafe-corner-sign}.  Thus \(\Scal_{h_a}\ge-C(h)\)
in the smoothing collar, uniformly as \(a\downarrow0\).
Choose a smooth \(c_a\ge0\), bounded by a constant depending
on the fixed metric, supported in \(|t|<4a\), and majorizing
the negative scalar part created in that collar.

Each orientable component of \(S\) bounds a compact handlebody.
Attach such caps, extend the metric smoothly through \(S\), and
denote their union by \(O\).  This gives a smooth complete
one-ended AF manifold on which to solve the correction.  Its
homogeneous Sobolev inequality
\begin{equation}
 \|v\|_{L^6}\le C\|\nabla v\|_{L^2},
       \qquad v\in C_c^\infty(X),
 \label{eq:unsafe-af-sobolev}
\end{equation}
has a constant uniform for small \(a\).  Indeed all these metrics
are uniformly equivalent to one fixed smooth AF metric.  The
Euclidean Sobolev inequality on the end and compact local Sobolev
inequalities leave only a compact \(L^2\) term.  Integration
along radial rays, followed by Cauchy--Schwarz with the integrable
weight \(r^{-2}\), controls the trace at a fixed end sphere by
the end gradient energy.  Poincar\'e's inequality with this trace
controls the remaining compact term, proving
\eqref{eq:unsafe-af-sobolev}.

Solve in the completion for the gradient norm
\begin{equation}
 -8\Delta_{h_a}u_a=c_a(1+u_a).
 \label{eq:unsafe-conformal-correction}
\end{equation}
Since \(\|c_a\|_{3/2}\to0\), the bilinear form
\(8\int|\nabla v|^2-\int c_av^2\) is coercive by
\eqref{eq:unsafe-af-sobolev}.  The right side is a bounded
functional with norm tending to zero, because
\(\|c_a\|_{6/5}\to0\).  Lax--Milgram gives a solution with
\(\|u_a\|_6+\|\nabla u_a\|_2=o(1)\).  Testing with its
negative part and using coercivity proves \(u_a\ge0\).
Local boundedness for scalar divergence equations, treating the
bounded \(c_a u_a\) term as a zeroth-order coefficient and the
\(L^2\)-small \(c_a\) as forcing, gives
\(\|u_a\|_\infty=o(1)\) on a fixed compact set.  On the
unchanged end, \(u_a\) is harmonic and tends to zero: this
follows first from its \(L^6\) membership and scaled local
estimates.  Exterior comparison then gives the global supremum
bound.  Elliptic estimates imply smooth convergence to zero on
compact sets off the seam.  The same end barriers as before give
\(u_a=O_2(|x'|^{-\beta})\).

Set \(h'=(1+u_a)^4h_a\) and restrict to the original side of
the caps.  The conformal scalar formula is
\begin{equation}
 \Scal_{h'}=(1+u_a)^{-5}
       \big(-8\Delta_{h_a}(1+u_a)+\Scal_{h_a}(1+u_a)\big)
       =(1+u_a)^{-4}(\Scal_{h_a}+c_a).
 \label{eq:unsafe-corrected-scalar}
\end{equation}
It is nonnegative off the unsafe set.  On that set the metric
was not mollified, \(c_a=0\), and \(\Scal_h\ge-1\);
since \(u_a\ge0\), the lower bound remains at least \(-1\).
In particular the claimed fixed lower bound \(-2\) holds.
The end scalar decay is preserved by the conformal formula.

The smoothing itself changes no asymptotic metric.  The conformal
ADM flux, followed by integration of
\eqref{eq:unsafe-conformal-correction} on the capped manifold,
gives
\begin{equation}
 m_{h'}-m_h
   =-\frac1{2\pi}\lim_{R\to\infty}
                   \int_{|x'|=R}\partial_nu_a\,\dd A_{h_a}
   =\frac1{16\pi}\int_X c_a(1+u_a)\,\dd V_{h_a}=o(1).
 \label{eq:unsafe-smoothing-mass}
\end{equation}
The decay \(\beta>1/2\) justifies replacing conormal-area
products in this flux as well.  Uniform metric convergence gives
the quadratic-form comparison.  Finally \(H_h(S)<0\) by
\eqref{eq:compact-mean-curvature}; the smooth convergence of the
correction near this disjoint compact surface preserves that
strict inequality.  Each smallness requirement here is imposed
after the fixed solution has been selected.
\end{proof}

\subsection{Enclosures whose boundaries avoid the unsafe set}

\begin{lemma}[Free enclosing boundaries]
\label{lem:free-enclosure}
Choose a sufficiently large fixed scaled radius \(R_*\), then
\(M\) sufficiently large depending on that radius, and perform
the smoothing of Lemma~\ref{lem:corner-smoothing} sufficiently
closely.  There is a perimeter minimizing enclosure \(E\) of
\(O\) and the marked balls about all points of \(\mathcal U_M\),
whose boundary is smooth, free, and disjoint from those balls and
from \(S\).  There is also a largest perimeter minimizing
enclosure \(E_0\) of \(O\) alone.  These sets have all the
boundary, exterior, and minimizing properties in
Proposition~\ref{prop:af-comparison}.
\end{lemma}

\begin{proof}
Extend the smooth inner base geometry across \(S\) in a fixed
collar for the purpose of defining balls.  Mark every closed base
ball
\begin{equation}
 \overline B_{g_{\rm ext}}(P,R_*r_P),\qquad P\in\mathcal U_M,
 \label{eq:unsafe-marked-balls}
\end{equation}
and include their union together with \(O\) as an obstacle.
For fixed parameters the union is compact: the center set is
compact and its positive radius function is continuous.  By
Lemma~\ref{lem:unsafe-blowup}, for each fixed \(R_*\) these
balls and the fixed larger multiples used below lie off the seam
once \(M\) is sufficiently large.  All lie inside the comparator
\(\mathcal S_1\).

Minimize \(h'\)-perimeter among precompact finite-perimeter
sets containing this obstacle modulo null sets.  A minimizer
exists by BV compactness and lower semicontinuity inside a
sufficiently large smooth truncation.  This is also a global
minimizer in the stated class.  To see that the outer truncation
causes no restriction, all sufficiently far AF spheres have
strictly positive outward mean curvature.  The unit normal field
of their foliation has positive divergence.  For almost every
cutting radius, Gauss--Green on the removed part of a competitor
shows that cutting inward saves perimeter if the removed volume
is positive.  The same argument excludes contact with the outer
constraint.  The fixed comparator and the small metric change
give a uniform bound, after enlarging \(C_*\) once,
\begin{equation}
 \Per_{h'}(E)\le C_*.
 \label{eq:unsafe-enclosure-bound}
\end{equation}

We show that its perimeter support cannot meet any marked ball.
Suppose it meets the ball marked at \(P\), at a point \(q\).
Use scaled smooth base coordinates with length unit \(r_P\)
on a fixed multiple of radius \(R_*\).  The base metric and
the comparison metric are equivalent to Euclidean metrics with
constants independent of \(R_*\), by
Lemma~\ref{lem:unsafe-blowup} and the chosen small smoothing.
If this region reaches \(S\), reflect in a base collar for these
comparisons.  Mixed entries of the graph metric vanish on the
face.  No comparison for the arbitrary cap metric is needed:
perimeters of sets containing \(O\) are independent of its
interior extension, and a reflected continuous metric can be
used there in the comparison argument.

The complement of \(E\) has uniform lower volume density at
\(q\) in these scaled coordinates.  Indeed filling a coordinate
ball preserves the obstacle condition.  Minimality and slicing
therefore bound its interior perimeter by a constant times the
complement's trace area on the ball.  If
\(V(a)=|B_a(q)\setminus E|\), Euclidean isoperimetry gives
\begin{equation}
 V(a)^{2/3}\le C V'(a)
 \quad\hbox{for almost every }a,\qquad
 V(a)\ge ca^3.
 \label{eq:unsafe-complement-density}
\end{equation}
The second inequality follows by integrating the first from zero;
the volume is positive at every radius because \(q\) is in
the perimeter support.  Apply this on a ball of radius comparable
to \(R_*\).  On a slightly larger common coordinate ball,
\(E\) itself has volume at least \(cR_*^3\), since it contains
the whole marked base ball modulo null sets.  The uniform
comparison of the scaled base metric with Euclidean metric
justifies this statement even when the marked ball crosses
\(S\).  Relative isoperimetry on that common ball now gives
\begin{equation}
 \Per_{h'}(E)\ge cR_*^2.
 \label{eq:unsafe-obstacle-perimeter}
\end{equation}
Choose \(R_*\) so that this exceeds \(C_*\), and only then
choose \(M\) large enough for all the scaled charts and their
fixed multiples.  This contradicts
\eqref{eq:unsafe-enclosure-bound}.  Compactness of the center set
also covers contacts with the closure of the union of marked balls.
This is where the radius-independent constant in
\eqref{eq:unsafe-quantifiers} is essential.

There is no contact with \(S\) either.  The parallel outward
normal field in a sufficiently thin collar of \(O\) has strictly
negative divergence, since \(H_{h'}(S)<0\).  Adjoin that collar
to a purported minimizer.  Gauss--Green on the added portion
bounds the newly exposed leaf area minus the removed perimeter
by the integral of this negative divergence.  The added volume
is positive for every sufficiently small width at a contact
point; otherwise that point would not be in the perimeter
support.  Slicing allows an almost every width for which all
perimeter formulas hold.  The resulting strict decrease
contradicts minimality.  Adjoining a collar is permitted whether
or not marked balls are present.

The perimeter support is thus separated from both obstacles and
the outer constraint.  It is locally perimeter minimizing in a
smooth three-manifold.  Interior codimension-one minimizing
boundary regularity gives a smooth compact embedded minimal
surface.  Take its open set representative for \(E\).  Filling
any bounded open exterior component would strictly reduce
perimeter, so its open exterior is connected.  Since every
precompact enlargement is an admissible competitor, \(E\) is
outward minimizing.  Its boundary is disjoint from every marked
ball, so the unsafe set lies in its interior.  The metric on
its closed exterior has nonnegative scalar curvature, and its
completeness and unique AF end follow from smoothness of compact
truncations and the AF estimates.

Now minimize with obstacle \(O\) alone.  The same compactness,
outer barriers, and strict inner mean-curvature barrier give
a smooth free minimal boundary and a connected open exterior.
Every component of this open minimizer meets \(O\), because
an open component disjoint from \(O\) can be removed, strictly
reducing perimeter.  Choose a minimizer of maximal volume within
the fixed truncation.  It is a largest minimizer modulo null
sets: the perimeter submodularity inequality implies that the
union and intersection of any two minimizers are minimizers,
and maximal volume forces the union to agree with this one.
Denote its open representative by \(E_0\).  Any proper
precompact enlargement of equal perimeter would also be a
minimizer and would contradict maximal volume.  The outer
barriers allow this reasoning without the truncation.  Thus
\(E_0\) is strictly outward minimizing.  Its boundary avoids
\(S\), so \(O\Subset E_0\).  Finally \(E\) is itself an
admissible enclosure of \(O\), proving
\eqref{eq:unsafe-two-perimeters}.
\end{proof}

No bound on the number or topology of components created by the
marked balls has been used.  In particular the subsequent area
transfer starts from \(E_0\), not from a topological assertion
about \(E\).  The hypotheses for the AF Riemannian Penrose theorem
are satisfied by the closed exterior of \(E\): it is smooth,
complete and one-ended, its scalar curvature is nonnegative and
decays as \(O(|x'|^{-2-2\beta})\), with \(2+2\beta>3\),
its metric decay has exponent \(\beta>1/2\),
and its entire possibly disconnected minimal boundary is outer
minimizing.  The boundary version of \citet[Theorem~1.4]{BrayLee2009}
therefore gives
\begin{equation}
 m_{h'}\ge\sqrt{\frac{\Per_{h'}(E)}{16\pi}}.
 \label{eq:unsafe-af-penrose}
\end{equation}

\subsection{Bad volume and completion of the comparison}

We finish the proof of Proposition~\ref{prop:af-comparison} by
checking the uniform quantitative inputs for
Proposition~\ref{prop:area-transfer}.  The exterior capacity
comparison in Section~\ref{sec:compact-construction} gives
\(x\ge1-\psi_\infty\), so for fixed \(\delta>0\) the set
\(\{x<1-\delta\}\) does not reach beyond a fixed outer sphere.
On the designated region, \eqref{eq:compact-bad-volume} gives
\(\int_{\{x<1-\delta\}}W\le C/L\).  The omitted exterior
collar has \(W=1\), so its contribution is bounded by its base
volume.  The omitted inner collar has base volume \(O(M^{-4})\).
Cauchy--Schwarz and \eqref{eq:compact-height-energy} together
with \eqref{eq:compact-bounds} therefore give
\begin{equation}
 \int_{\{x<1-\delta\}}W\,\dd V_{\rm base}
 \le \frac CL+\Vol_{g_o}(\text{omitted exterior collar})
       +C(L)M^{-1}.
 \label{eq:unsafe-total-bad-volume}
\end{equation}
The coefficient in \(C/L\) is independent of \(L\).  The
other constants may depend on the earlier collar choice and
on \(L\), which is harmless: first make the exterior collar
thin and \(L\) large, then let \(M\) be large.

Choose \(b_*,\delta_0\) small enough in
\eqref{eq:unsafe-mass-limsup} for the prescribed mass tolerance.
For the prescribed \(\delta\) and volume tolerance choose the
collar and \(L\) as just described.  Fix the common comparator
bound, choose \(R_*\) in Lemma~\ref{lem:free-enclosure}, and
then choose \(M\) large enough for that lemma, the scalar
lower bound, the mass costs, and
\eqref{eq:unsafe-total-bad-volume}.  The subsequent small lapse
cutoff and finite strengths are those of
Proposition~\ref{prop:compact-solutions}; all uniform estimates
above hold for those choices.  Apply
Lemma~\ref{lem:corner-smoothing} with mass and metric errors
small enough for the remaining tolerances.  These choices prove
\eqref{eq:unsafe-comparison-output}--\eqref{eq:unsafe-bad-volume-output}.

The caps may be fixed once as topological handlebodies, regardless
of all metric parameters.  The topology entering the area-transfer
argument is consequently the fixed finite topology of
\(X\setminus\operatorname{int}O\).  Positive-time flow levels
outside \(E_0\) are allowed to enter the unsafe region: the
uniform lower scalar bound there is precisely the input supplied
by \eqref{eq:unsafe-comparison-output}.  The mass theorem is
applied to the unmodified smooth exterior of \(E\), whereas
the small initial mean-curvature change used solely for the flow
is covered by Proposition~\ref{prop:area-transfer}.  This
completes the AF comparison construction.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/09-scalar-error.tex ====

==== BEGIN The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/10-completion.tex ====
\section{Completion of the proof}
\label{sec:completion}

\subsection{The time-symmetric inequality}

\begin{proof}[Proof of Theorem~\ref{thm:ts}]
The approximation of Proposition~\ref{prop:conformal-approximation}
preserves $H(S)\le0$, converges uniformly in metric comparison, and
preserves the limiting time component. The comparison of enclosing
areas in Section~\ref{sec:preliminaries} therefore allows us first to
assume that $R_g+6>0$ and that the end has the smooth conformal form
used in Section~\ref{sec:stress-extension}. Fix this approximated
metric for the constructions that follow, and put
\[
 A_*=A_{\min}(S;g),\qquad a=\sqrt{A_*/(4\pi)}.
\]

Given an arbitrarily small mass tolerance $\varepsilon_m>0$,
Proposition~\ref{prop:af-attachment} produces a fixed compact inner
piece, a smooth synthetic tensor on it, and a scalar-flat AF
attachment with mass $m_o$ satisfying
\begin{equation}\label{eq:completion-attachment-budget}
 m_o\le p_0(g)-\frac{a^3}{2}+\varepsilon_m.
\end{equation}
The electrostatic height, attachment radius, and smoothing tolerances
used to obtain this inequality have already been fixed. They are
independent of the large graph height in the next construction.

Apply Proposition~\ref{prop:compact-solutions} and the comparison
construction of Proposition~\ref{prop:af-comparison}. Their parameters
are chosen in the proved order: the small inner flux and scalar
cutoff costs first, then a desired relative-area tolerance
$0<\varepsilon_A<1$, a sufficiently small contraction threshold,
a thin omitted exterior collar, and a sufficiently large fixed
volume penalty; next the graph height is taken large, and finally
the lapse clipping, finite penalty strengths, and corner-smoothing
tolerances are chosen. The costs can be made small enough that the
resulting smooth AF comparison metric satisfies
\begin{equation}\label{eq:completion-comparison-budget}
 m_{\AF}(h')\le m_o+\varepsilon_m.
\end{equation}

Let $E$ be its minimizing enclosure containing the caps and the
unsafe-region obstacles. Its free boundary is smooth, compact,
minimal, and outer minimizing. The exterior is complete and has
$R_{h'}\ge0$, the AF decay required in Theorem~\ref{thm:af-penrose},
and one end. Let $E_0$ instead be a largest perimeter-minimizing
enclosure of the caps alone. Then
\begin{equation}\label{eq:completion-enclosure-order}
 \Area_{h'}(\partial E)\ge\Area_{h'}(\partial E_0).
\end{equation}
The exterior of $E_0$ has a scalar lower bound independent of the
large graph height, even where it meets the unsafe region.
The bad-volume estimates from Sections~\ref{sec:compact-construction}
and~\ref{sec:scalar-error}, together with the projection along the
AF attachment, satisfy Proposition~\ref{prop:area-transfer}.
Consequently, when $A_*>0$, the parameters above can be chosen so that
\begin{equation}\label{eq:completion-area}
 \Area_{h'}(\partial E_0)\ge(1-\varepsilon_A)A_*.
\end{equation}
The small initial conformal change used only for the flow fixes this
initial area. It is not made in applying the mass theorem.

Theorem~\ref{thm:af-penrose} applies to the exterior of $E$, not the
possibly unsafe exterior of $E_0$. Equations
\eqref{eq:completion-attachment-budget}--\eqref{eq:completion-area}
give
\begin{align*}
 p_0(g)
 &\ge \frac{a^3}{2}+m_o-\varepsilon_m\\
 &\ge \frac{a^3}{2}+m_{\AF}(h')-2\varepsilon_m\\
 &\ge \frac{a^3}{2}+\frac a2\sqrt{1-\varepsilon_A}
                         -2\varepsilon_m.
\end{align*}
Let the two tolerances tend to zero, making fresh choices of the
finite parameters as necessary. If $A_*=0$, the same construction
and the nonnegativity consequence of Theorem~\ref{thm:af-penrose}
give $p_0(g)\ge-2\varepsilon_m$ without the area step.
Thus \eqref{eq:ts-inequality} holds for every fixed approximated
metric. Passing back through the preliminary approximation proves
it for the original metric.
\end{proof}

\subsection{Maximal data and invariant mass}

\begin{proof}[Proof of the inequality in Theorem~\ref{thm:main}]
The maximal constraint and \eqref{eq:dec-integrability} imply
\[
 \int_{\mathrm{end}} V_0|R_g+6|\dd V_g<\infty.
\]
Use Proposition~\ref{prop:mass-covariance} to choose a rest chart
for the original covector. All hypotheses remain valid in that
chart and $p_0(g)=m_{\AH}(g)$.

Define
\[
 F(A)=\frac12\left(\sqrt{\frac A{4\pi}}
                       +\left(\frac A{4\pi}\right)^{3/2}\right).
\]
This function is continuous, nonnegative, and nondecreasing on
$[0,\infty)$. Theorem~\ref{thm:ts} supplies the time-symmetric input
of Theorem~\ref{thm:ah-reduction}, including comparison metrics whose
covectors are not timelike. The reduction therefore gives
\[
 p_0(g)\ge F\bigl(A_{\min}(S;g)\bigr).
\]
Since the chart was fixed before any construction, this is precisely
\eqref{eq:main-inequality}. The one-sided time-component budgets
suffice; there is no further assertion about the spatial components
of intermediate comparison metrics.
\end{proof}

\subsection{Sharpness}

Let $m>0$, and let $a$ be the positive solution of
$a+a^3=2m$. The Schwarzschild--anti-de Sitter metric is
\begin{equation}\label{eq:sads-metric}
 g_m=\frac{\dd r^2}{1+r^2-2m/r}+r^2\sigma,
 \qquad r\ge a,\qquad K=0.
\end{equation}
To include its boundary smoothly, use proper distance $s$ from
$r=a$. Then
\[
 g_m=\dd s^2+r(s)^2\sigma,
 \quad r(0)=a,\quad r'(0)=0,
 \quad (r')^2=1+r^2-2m/r.
\]
The simple positive radial derivative of the expression on the
right at $a$ gives a smooth even solution $r(s)$ near zero, increasing
for $s>0$. Hence the horizon is a smooth minimal boundary and the
exterior is complete with that boundary included. Since
$r''=r+m/r^2$, the warped-product formula gives
\[
 R_{g_m}=-\frac{4r''}{r}
             +\frac{2(1-(r')^2)}{r^2}=-6.
\]
The constraints are vacuum, the metric error and its required tensor
derivatives are $O(r^{-3})$, and the weighted matter integral vanishes.

For completeness the mass normalization can be checked directly.
In the background distance coordinate, write
$g_m-b=\zeta\dd\eta^2$, where
\[
 \zeta=\frac{2m}{r(1+r^2-2m/r)}.
\]
The radial component of $\divg_b e-\dd\tr_be$ is
$2\coth\eta\,\zeta$, and the other two terms of
\eqref{eq:mass-integrand} cancel for $V_0$.
The normalized sphere flux therefore tends to
\[
 m\frac{1+r^2}{1+r^2-2m/r}\longrightarrow m.
\]
Rotational symmetry makes the three spatial fluxes zero, so
$m_{\AH}(g_m)=m$.

Radial projection onto the horizon has two-area Jacobian at most one,
since $r\ge a$. Every radial ray meets the entire intrinsic boundary
of any admissible end-side domain: either it enters that domain
across its boundary, or its initial point already lies on the counted
boundary at $S$. Thus projection of the cut covers the horizon.
The area formula gives $\Area_{g_m}(\partial D)\ge4\pi a^2$.
The horizon itself is an admissible competitor, and hence
$A_{\min}(S;g_m)=4\pi a^2$. Equation $2m=a+a^3$ proves equality in
\eqref{eq:main-inequality} and completes Theorem~\ref{thm:main}.
==== END The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data//sections/10-completion.tex ====

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