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\begin{document}
\title{Conformal flow and the Riemannian Penrose inequality\\ with minimizing frontiers}
\author{A self-contained account}
\date{2026}
\maketitle
\begin{abstract}
A black hole stores its mass in the geometry of its horizon. The Penrose
inequality is the sharp statement that a given horizon area forces a lower
bound on total mass. In its Riemannian (time-symmetric) form it reads
$m\ge\tfrac12(A/\omega)^{(n-2)/(n-1)}$, where $m$ is the ADM mass, $A$ the
total horizon area, and $\omega$ the area of the unit $(n-1)$-sphere. This
note proves the inequality in every dimension $n\ge3$ for complete
asymptotically flat manifolds of nonnegative scalar curvature whose horizon
is an outer-minimizing, locally area-minimizing boundary. The engine is the
conformal flow: a family of metrics that keeps the horizon area fixed while
driving the mass monotonically down to the sharp bound. We follow Bray's
conformal flow as developed by Bray--Lee and Bi--Zhu, and we present each
analytic ingredient (area preservation, the mass--capacity comparison,
separation of singular frontiers, and the final limit) in one place with its
proof.
\end{abstract}
\section{Introduction}
\subsection{Mass, horizons, and the inequality}
General relativity ties mass to geometry. Consider a moment of time symmetry,
when the spatial slice carries no momentum. The spacetime geometry then
reduces to a Riemannian manifold $(M,h)$, and the energy condition that
``density is nonnegative'' becomes the single geometric hypothesis
\begin{equation}\label{eq:intro-rsc}
R_h\ge0,
\end{equation}
nonnegative scalar curvature. An apparent horizon, the surface from which
light cannot escape, becomes a minimal surface $\Sigma$: a hypersurface whose
mean curvature, measured with the normal that points toward spatial infinity,
vanishes.
Penrose's $1973$ argument \cite{Penrose1973} combined three physical facts. Cosmic censorship
forbids the horizon from being seen; the area of a horizon never decreases;
and a collapsed object settles into a Kerr black hole. From an initial trapped
area these force a lower bound on the total mass remaining after collapse. The
Riemannian problem keeps the mass--area comparison and discards the
spacetime-evolution hypotheses. It asks only: how much mass must a
time-symmetric slice carry if it contains a horizon of a given size?
Write $n$ for the dimension, $k=n-1$ for the dimension of the horizon, $d=n-2$,
and $\omega=\mathcal H^k_\delta(S^k)$ for the area of the unit $k$-sphere. The
numerical inequality is
\begin{equation}\label{eq:intro-rpi}
m_{\rm ADM}(h)\;\ge\;\frac12\left(\frac{A}{\omega}\right)^{d/k},
\end{equation}
where $m_{\rm ADM}$ is the ADM mass measured at spatial infinity and $A$ is the
total horizon area, counting every component and every portion of the frontier.
\subsection{The main theorem}
We work with an exterior: a complete manifold that contains exactly one
Euclidean coordinate end and may terminate at an inner boundary. The frontier
$\Sigma$ may be a single smooth surface or a singular finite-perimeter set. To
count its area correctly we record the side that is ``filled'' (the black-hole
side). The area $A$ is then the full perimeter of that filled side.
\begin{theorem}[Riemannian Penrose inequality]\label{thm:intro-main}
Let $(E,h)$ be a connected complete $n$-dimensional exterior, $n\ge3$, with
compact frontier $\Sigma$, and suppose $h$ is smooth on $E$ and through
$\Sigma$. Assume
\begin{enumerate}
\item $R_h\ge0$, $R_h\in L^1(E,dV_h)$, and $R_h=O(r^{-n-\beta})$ for some
$\beta>0$ on the end;
\item outside a compact set there is exactly one Euclidean coordinate end with
$h-\delta=O_2(r^{-\tau})$, where $\tau>d/2$;
\item the full frontier is outer-minimizing and locally perimeter-minimizing as
the boundary of the filled side; its regular part is a smooth embedded
minimal hypersurface, and its singular set has Hausdorff dimension at
most $n-8$.
\end{enumerate}
Then
\begin{equation}\label{eq:intro-main}
m_{\rm ADM}(h)\ge\frac12\left(\frac{\mathcal H_h^k(\Sigma)}{\omega}\right)^{d/k}.
\end{equation}
\end{theorem}
All boundary components enter the area. Neither spin nor connectedness of the
frontier is assumed. The zero-area case is included: the density lower bound
makes a nonempty locally minimizing frontier have positive perimeter, so zero
area means an empty frontier, and the conclusion is simply $m_{\rm ADM}\ge0$.
The hypotheses say three things in words. Nonnegative scalar curvature is the
energy condition. The one-end condition fixes the mass at a single infinity.
The ``outer-minimizing, locally minimizing'' condition on the frontier says the
horizon is stable: it cannot be pushed inward or outward to reduce area, and it
is the outermost such surface. That last stability is exactly what makes the
conformal flow well behaved.
\subsection{Sharpness: the Schwarzschild--Tangherlini example}
The constant in \eqref{eq:intro-main} cannot be improved. The
Schwarzschild--Tangherlini metric \cite{Tangherlini1963}
\begin{equation}\label{eq:intro-schwarzschild}
h_m=\left(1+\frac{m}{2r^d}\right)^{4/d}\delta,
\qquad r\ge\left(\frac m2\right)^{1/d},\quad m>0,
\end{equation}
has nonnegative scalar curvature, is asymptotically flat, and has a minimal
inner sphere at $r=(m/2)^{1/d}$. Its ADM mass is $m$, its horizon area is
$\omega(2m)^{k/d}$, and a direct substitution shows that it meets
\eqref{eq:intro-main} with equality. Thus the inequality is sharp and, at
equality, the Schwarzschild--Tangherlini family is the expected model.
\subsection{Brief history}
The three-dimensional case was settled in two independent ways. Huisken and
Ilmanen's weak inverse mean curvature flow \cite{HuiskenIlmanen2001} controls
the Hawking mass along a monotone family of surfaces and proves the sharp
bound for each connected component of an outermost minimal boundary. Bray's
conformal flow \cite{Bray2001} changes the ambient metric while keeping the
total horizon area fixed, and thereby proves the bound for the full area even
when the boundary is disconnected. The two methods agree on the constant but
neither contains the other.
The passage to other dimensions is where singularities enter. Bray and Lee
\cite{BrayLee2009} extended the conformal method to dimensions $3\le n\le7$,
where a minimizing area boundary is smooth. In dimensions $n\ge8$ such a
frontier may be singular, and its singular points disturb the flow, the
separation of successive slices, and the geometric comparison that makes mass
decrease. Bi and Zhu \cite{BiZhu2026v2} proved the inequality in all dimensions
together with Schwarzschild rigidity for almost-minimizing finite-perimeter
frontiers that are their own outermost minimizing enclosures. Restricted
all-dimensional results for graphical hypersurfaces came earlier
\cite{Lam2010Graphs,deLimaGirao2015}, using extra graphical control.
This note proves the numerical inequality \eqref{eq:intro-main}, not the
rigidity statement, in every dimension. We follow the Bi--Zhu conformal flow
but require the frontier to be locally perimeter-minimizing on both sides with
a metric that is smooth through it. The plan is to show which geometric and
positive-mass inputs actually suffice, with every step in one place.
\subsection{Strategy of the proof}
Assume first that the initial metric $g_0=h$ has a harmonically flat end, so
that $g_0=H^{4/d}\delta$ there with $H>0$ Euclidean harmonic and $H\to1$. (The
last section removes this convenience by a cutoff and a Dirichlet correction.)
We construct a one-parameter family of metrics $g_t=u_t^{4/d}g_0$ with filled
regions $K_t$ whose frontiers $\Sigma_t$ keep the area $A$ fixed and whose ADM
mass decreases to a limit. Four quantities run the proof:
\[
A=P_{g_t}(K_t),\quad
m(t)=m_{\rm ADM}(g_t),\quad
\mu(t)=m(t)-c_t,\quad
R_t\ \text{(outer radius)}.
\]
Here $c_t=\mathfrak c(\Sigma_t,g_t)$ is the capacity of the conductor, defined
by \eqref{eq:rpi:capacity} below: the normalized Dirichlet energy of the
harmonic function equal to one on the frontier and zero at infinity.
The logic is a chain of comparisons.
\begin{enumerate}
\item The frontier is replaced by its outermost minimizing enclosure
(Section~\ref{sec:g-enclosures}); this preserves the area and makes the
horizon stable.
\item The conformal flow is built from harmonic velocity potentials, and it
preserves the area exactly (Section~\ref{sec:rpi:flow-construction}).
\item Successive flow slices are shown to be disjoint, including at singular
points, via regular estimates and a single distance median
(Section~\ref{sec:rpi:regular-patches}).
\item A smooth mass--capacity inequality $m\ge c$ follows from positive mass by
doubling the exterior and repairing the seam
(Sections~\ref{sec:rpi:orientation}--\ref{sec:rpi:flow-mass-capacity}).
This gives $\mu\ge0$ and, along the flow, $m'=-2\mu$.
\item The deficit satisfies $\mu(t)\to0$, and every normalized frontier stays
inside a fixed ball (Section~\ref{sec:rpi:flow-estimates}).
\item Complete smooth comparison metrics with vanishing mass reduce the end
factor to a radial harmonic function (Section~\ref{sec:rpi:lee-completion}).
\item The limiting radial potential identifies the largest frontier radius,
which yields the area bound \eqref{eq:intro-main}
(Section~\ref{sec:rpi:numerical-limit}).
\end{enumerate}
The order matters: area preservation is proved before the nesting of limiting
hulls, and the frontier is separated from the singular set before capacity is
compared to mass. Section~\ref{sec:rpi:density-repair} removes the harmonic
flatness assumption at the end without disturbing any enclosure area.
\subsection{Conventions}
We use $O_j(r^{-\gamma})$ for a tensor whose coordinate derivatives of order
$\ell\le j$ are $O(r^{-\gamma-\ell})$. On a Euclidean coordinate end the ADM
mass is
\begin{equation}\label{eq:intro-adm}
m_{\rm ADM}(h)=\frac1{2k\omega}\lim_{R\to\infty}\int_{S_R}
(\partial_jh_{ij}-\partial_ih_{jj})\,n_\delta^i\,dA_\delta .
\end{equation}
The decay and scalar integrability hypotheses of Theorem~\ref{thm:intro-main}
make this limit well defined \cite{Bartnik1986}.
For a frontier bounding a filled region, take the normal $\nu$ that points
toward the asymptotic end, and write $H_\nu=\divg_\Sigma\nu$. The outward
normal of the exterior region is then $-\nu$. Every mean-curvature sign below
is stated using one of these two explicit normals, so no confusion between
conventions is possible.
\section{Enclosing area and minimizing frontiers}
\label{sec:g-enclosures}
The horizon in Theorem~\ref{thm:intro-main} is only assumed to be an
outer-minimizing boundary. A convenient object to flow is the minimizing
enclosure of the filled region: the smallest set of that class. This
section shows that such an enclosure exists, that its frontier is
regular, and that a strictly mean-concave inner barrier forces it away
from the boundary. None of this uses nonnegative scalar curvature; it is
pure perimeter geometry, and it is exactly the input the flow needs.
\subsection{Full cuts and the full area}
Let $(X,g)$ be a smooth connected Riemannian $n$-manifold, $n\ge3$,
complete including its nonempty compact smooth boundary $B$, with one
coordinate end and compact end complement. In the end coordinates let
\begin{equation}\label{eq:g-end}
g_{ij}-\delta_{ij}=O_2(r^{-q})\qquad\text{for some }q>0.
\end{equation}
A \emph{full smooth enclosing cut} is the complete intrinsic boundary
$\partial_{\mathrm{man}}D$ of a connected smooth domain $D\subset X$ that
is closed in $X$, contains the distant coordinate end, and has compact
boundary lying in $\operatorname{int}X$. Every boundary component and
every portion coincident with $B$ is counted. In particular $D=X$ is
admissible and its cut is $B$. Define
\begin{equation}\label{eq:g-cutarea}
a_g(B)=\inf_D\Area_g(\partial_{\mathrm{man}}D).
\end{equation}
To make area minimization well posed we work on an enlarged manifold.
Attach a compact smooth filling $O$ behind $B$ and extend $g$ smoothly
across it, producing a boundaryless manifold $M$ with
$M\setminus O=\operatorname{int}X$. The extension is auxiliary: the
enclosure lemma below shows it does not change \eqref{eq:g-cutarea}.
Let $\mathcal A$ be the bounded finite-perimeter subsets $E\subset M$
that contain $O$ up to a null set, with sets identified when they agree
almost everywhere. Their full ambient perimeter is
\begin{equation}\label{eq:g-perimeter}
P_g(E)=|D\mathbf 1_E|_g(M)=\mathcal H^{n-1}_g(\partial^*E).
\end{equation}
This counts $P_g(O)=\Area_g(B)$, whereas a perimeter relative to
$\operatorname{int}X$ would assign $O$ zero area and lose all contact
with $B$.
\subsection{The full enclosure minimum}
\begin{lemma}[Full enclosure]\label{lem:g-enclosure}
Under the preceding hypotheses,
\begin{equation}\label{eq:g-minimum}
0<a_g(B)=\min_{E\in\mathcal A}P_g(E)\le\Area_g(B).
\end{equation}
All minimizers are supported in one compact set and form a compact family
for $L^1(M)$ convergence, which also carries their perimeters. Each
minimizer has a closed representative $E$ with frontier
$T=\operatorname{supp}|D\mathbf 1_E|_g$, whose exterior $M\setminus E$ is
connected and contains the distant end. Full smooth enclosing cuts have
areas tending to $P_g(E)$, and for $3\le n\le7$ they can be chosen
$C^1$-convergent to $T$. Every minimizer is outer minimizing: if
$F\supset E$ is bounded with finite perimeter, then $P_g(F)\ge P_g(E)$.
\end{lemma}
\begin{proof}
\emph{Confinement.} On the end put $Y=\nabla r/|\nabla r|_g$. Its
divergence is the outward mean curvature of the coordinate spheres, so
\eqref{eq:g-end} gives $\divg_gY=(n-1)/r+O(r^{-1-q})>0$ for $r>R_0$.
For $E\in\mathcal A$ and almost every $R>R_0$, Gauss--Green on
$E\cap\{r>R\}$ (the trace exists and the sphere carries no mass) yields
\[
\int_{E\cap\{r>R\}}\divg_gY\,dV_g
=\int_{\partial^*E\cap\{r>R\}}g(Y,\nu_E)\,dA_g-\int_{S_R}\mathbf1_E\,dA_g .
\]
The first boundary term is at most the removed perimeter and the second
is the sphere area added by clipping. Hence the truncated set
$E_R=E\setminus\{r>R\}$ has $P_g(E_R)\le P_g(E)$, strictly if the
removed tail has positive volume. Minimizing over the bounded truncation
at a fixed $R_1>R_0$ (the competitor $O$ bounds the perimeter) gives a
limit by BV compactness and lower semicontinuity. Strictness of clipping
confines every minimizer to the truncation and shows the constrained
minimum is global. The minimum is positive: a zero-perimeter indicator
would be constant on connected $M$, while every admissible set contains
the nonempty open $O^\circ$ and misses an end tail. Compactness of
minimizers is the same BV argument, since the limit stays admissible and
lower semicontinuity plus minimality forces equal perimeter.
\emph{Recovery by smooth cuts.} A smooth exterior $D$ gives the filled
competitor $E_D=O\cup(X\setminus\operatorname{int}D)$ with
$P_g(E_D)=\Area_g(\partial_{\mathrm{man}}D)$, so the perimeter infimum is
at most $a_g(B)$. Conversely, start from any $E\in\mathcal A$, flow $O$
slightly along a collar into $X$ to get $E_t\supset O$ with a positive
exterior collar and $P_g(E_t)\to P_g(E)$, and approximate $\mathbf1_{E_t}$
strictly in $BV$ by smooth $u_j\in[0,1]$ (constant one near $O$, zero
outside a fixed compact). Coarea and Sard give regular values $s_j$ with
$F_j=\{u_j>s_j\}$ containing a neighbourhood of $O$, compact smooth
boundary, and $P_g(F_j)\to P_g(E_t)$. Retaining the end component of
$M\setminus\overline{F_j}$ and filling its bounded complementary
components gives a connected smooth exterior $D_j$ with
\[
a_g(B)\le\Area_g(\partial_{\mathrm{man}}D_j)\le P_g(F_j).
\]
Letting $j\to\infty$ then $t\downarrow0$ proves $a_g(B)\le P_g(E)$ for
every competitor, hence \eqref{eq:g-minimum} and the cut approximations.
\emph{Connected exterior.} The regularity lemma below identifies a
closed representative with frontier $T$ whose singular part is
$\mathcal H^{n-1}_g$-null. A bounded complementary component $U$ has
$\partial U\subset T$ of finite measure, hence finite perimeter, and on
its regular charts the outer normal is opposite to $\nu_E$. The union
formula gives $P_g(E\cup U)=P_g(E)-P_g(U)<P_g(E)$, contradicting
minimality; so the exterior is connected, without assuming a smooth
frontier. For $n\le7$ the frontier is $C^{1,1}$; displacing it slightly
and smoothing a defining function in $C^1$ produce full cuts with areas
tending to $P_g(E)$.
Finally every bounded finite-perimeter superset of a minimizer still
contains $O$, so global minimality gives outer minimization directly.
\end{proof}
\subsection{Regularity at a smooth obstacle}
The connectedness argument above used a representative whose frontier has a
small singular set, and the detachment argument below will need contact with
$B$ to be regular. Both are consequences of a local statement about perimeter
minimization constrained by a smooth obstacle. It needs no global existence.
\begin{lemma}[Smooth-obstacle regularity]\label{lem:g-obstacle}
Let $M^n$ carry a smooth metric, let $O$ have smooth boundary, and let $E$
locally minimize perimeter among sets containing $O$. On a relatively compact
region meeting $\partial O$, assume $\partial O$ has bounded smooth geometry.
Then $E$ has a closed representative whose topological frontier is
$T=\operatorname{supp}|D\mathbf 1_E|_g$. Its regular part has multiplicity one,
and its singular set $\mathcal S$ satisfies
\[
\mathcal S=\varnothing\quad(n\le7),\qquad
\dim_{\mathrm H}\mathcal S\le n-8\quad(n\ge8).
\]
The regular part outside $\partial O$ is smooth and minimal. Every contact
point in $T\cap\partial O$ is regular, with local graph in
$C^{1,\alpha}\cap W^{2,p}$ for every $0<\alpha<1$ and every $1<p<\infty$. For
$3\le n\le7$ the entire frontier is $C^{1,1}$, with local graph functions in
$W^{2,\infty}$. Moreover $\mathcal H^{n-1}_g(\mathcal S)=0$, and
$\mathcal H^{n-1}_g(T)=P_g(E)$ wherever the total perimeter is finite.
\end{lemma}
\begin{proof}
\emph{From a constraint to a volume error.} Extend the outward unit normal of
$O$ to a smooth field $Z$ with $|Z|_g\le1$ and bounded divergence $\Lambda$,
cut off in a collar. For a local replacement $F$, the set $F\cup O$ is
admissible. Perimeter submodularity and Gauss--Green, cancelling perimeter
unchanged outside the replacement region $V$, give
\begin{equation}\label{eq:g-almost-minimality}
P_g(E;V)\le P_g(F;V)+\Lambda\Vol_g(E\triangle F).
\end{equation}
The obstacle has become an ordinary almost-minimization estimate.
\emph{Density and the regular part.} Insertion and deletion of small balls in
\eqref{eq:g-almost-minimality}, followed by the relative isoperimetric
inequality, give two-sided volume-density and perimeter-density bounds. These
identify the frontier of the closed representative with the support of its
perimeter measure. The codimension-one almost-minimizer regularity theorem
then gives a $C^{1,\alpha}$ regular part and a singular set of dimension at
most $n-8$. The error in a radius-$r$ ball is $O(r^n)$, so rescaled blowups
are genuine Euclidean perimeter-minimizing cones \cite{Maggi2012,SimonGMT}.
\emph{Contact points are regular.} Take a tangent filled set at a point of
$\partial O$. It contains the tangent half-space of $O$, so its boundary cone
lies in the complementary closed half-space and is a stationary
multiplicity-one minimizing cone. Write $k=n-1$. On its spherical link the
nonnegative coordinate height $z$ satisfies weakly
$\Delta z=-(k-1)z$; this is the restriction of linear functions to a
stationary cone. The link is compact and stationarity includes its singular
set, so testing with the constant function gives $0=-(k-1)\int z$. Since
$k-1>0$, $z$ vanishes, and the boundary cone is the bounding hyperplane of a
half-space. Constancy and the density bounds give multiplicity one, and
density-one regularity makes the contact a single graph.
\emph{Smoothness and dimensions.} Apply \eqref{eq:g-almost-minimality} to
flows of both signs in such a graph cylinder. The graph function $v$ then
solves a uniformly elliptic prescribed-mean-curvature equation with bounded
right-hand side. Difference quotients give $W^{2,2}$ regularity, and the
linear $W^{2,p}$ estimates in nondivergence form give $W^{2,p}$ for every
finite $p$; Sobolev embedding gives $C^{1,\alpha}$ for each $\alpha<1$
\cite{GilbargTrudinger}. Off the obstacle the weak mean curvature is zero, so
interior regularity makes every regular patch smooth and minimal.
For $n\le7$ apply the smooth-obstacle theorem of
\cite{HuiskenIlmanen2001}: a pure-perimeter minimizer against a $C^2$
obstacle in ambient dimension below eight has $C^{1,1}$ frontier, smooth away
from contact. Its graph functions lie in $W^{2,\infty}$. Finally the
dimension estimate makes the singular set $\mathcal H^{n-1}$-null, and the
regular frontier agrees with the reduced boundary up to such a null set,
proving the area identity.
\end{proof}
\subsection{Detachment from a strict inner barrier}
Contact with the obstacle is possible when $B$ has zero mean curvature. Strictly
negative mean curvature instead forces every minimizing enclosure to leave a
fixed collar, and the resulting frontier is an ordinary minimal surface in a
smooth neighbourhood.
\begin{lemma}[Detachment from a strict barrier]\label{lem:g-detachment}
In the setting of Lemma~\ref{lem:g-enclosure}, suppose
$H_B=\divg_B\nu_B<0$ on every component of $B$, where $\nu_B$ points into $X$.
There is $\varepsilon>0$, depending only on a fixed collar of $(B,g)$, such
that every minimizing filled set contains the exterior distance collar
$0\le t<\varepsilon$ in its interior. Consequently its frontier $T$ is compact,
nonempty, disjoint from $B$, and locally perimeter minimizing in the smooth
ambient metric. It is stationary as a whole boundary varifold, smooth and
minimal outside a compact singular set of dimension at most $n-8$, and a smooth
minimal full cut when $3\le n\le7$. The closed exterior
$D=\overline{M\setminus E}$ is connected, has the same single end, and is a
complete closed subset for the ambient distance. Its frontier is outer
minimizing with all components counted, and $\mathcal H^{n-1}_g(T)=a_g(B)$.
\end{lemma}
\begin{proof}
\emph{The collar is filled.} Compactness of $B$ and strict negativity give a
signed-distance collar $0\le t\le2\varepsilon$ on which $Y=\partial_t$ obeys
$|Y|_g=1$ and $\divg_gY\le-\beta<0$. For a good slicing level
$s\in(0,2\varepsilon)$ set $O_s=O\cup\{0<t<s\}$ and $W_s=O_s\setminus E$.
Gauss--Green gives
\begin{equation}\label{eq:g-inner-collar}
P_g(E\cup O_s)-P_g(E)
\le\int_{W_s}\divg_gY\,dV_g\le-\beta\Vol_g(W_s).
\end{equation}
The enlarged set is admissible, so minimality forces $\Vol_g(W_s)=0$. Choosing
one good level per minimizer, the common smaller collar $0\le t<\varepsilon$ is
filled almost everywhere; the density bounds of Lemma~\ref{lem:g-obstacle}
then put it in the interior of $E$.
\emph{Local minimality and stationarity.} The frontier is now a positive
distance from $B$, so every local perimeter replacement near it is admissible,
and it is locally perimeter minimizing. To see this in the language of
currents, take a coordinate ball $U\Subset M\setminus O$ and orient it. Write
$C=\partial[\![E]\!]$ as a current in $U$. If $Z$ is an integral
$(n-1)$-cycle compactly supported in $U$, coning it fills it by an integral
$n$-current $S=[\![k]\!]$ with $k\in BV(U)$ of compact support and $\partial
S=Z$. The $1$-Lipschitz truncation $c(t)=\min\{1,\max\{0,t\}\}$ maps integers
to $\{0,1\}$, so $c(\mathbf 1_E+k)=\mathbf 1_F$ for a finite-perimeter $F$
agreeing with $E$ outside a compact set. The contraction property and set
minimality give
\[
P_g(E;U)\le P_g(F;U)\le|D(\mathbf 1_E+k)|_g(U)=\mathbf M_g(C+Z).
\]
Applying ambient flows of both signs and summing gives
$\int_{\partial^*E}\divg_{T_x\partial^*E}V\,dA_g=0$ for every smooth $V$
supported near $T$: stationarity, with no extra term at the singular set.
Lemma~\ref{lem:g-obstacle} and interior minimizing-boundary regularity give the
stated dimensional conclusions \cite{SimonGMT}.
\emph{Completeness.} Lemma~\ref{lem:g-enclosure} gives a connected, bounded
removed region, so the exterior keeps exactly the original end and is closed in
the complete manifold $M$. For an intrinsic Cauchy sequence, choose a
subsequence with successive distances below $2^{-j}$ and join consecutive
terms by curves in $D$ of length below $2^{1-j}$; their concatenation has
finite length and an ambient endpoint, which lies in $D$, and the remaining
tail lengths vanish. So the sequence converges intrinsically. Outer
minimization and the full-area identity follow from \eqref{eq:g-minimum} and
the zero $(n-1)$-measure of the singular set.
\end{proof}
\subsection{The inequality for a strict inner barrier}
Combining the two lemmas with the main theorem gives a mass bound for a smooth
inner boundary that is strictly mean-concave toward the end, without assuming
the boundary itself minimizes area.
\begin{corollary}[Strict inner barrier]\label{cor:g-strict-barrier-penrose}
Let $(X,g)$ be a smooth connected $n$-dimensional exterior, $n\ge3$, complete
including its nonempty compact smooth boundary $B$, with one Euclidean
coordinate end and compact end complement. Suppose
\[
g-\delta=O_2(r^{-\tau}),\qquad \tau>\frac{n-2}{2},\qquad
R_g\ge0,\quad R_g\in L^1(X,dV_g),\quad R_g=O(r^{-n-\beta})\ (\beta>0).
\]
If $H_B<0$ for the normal pointing into $X$, then
\[
m_{\rm ADM}(g)\ge\frac12\left(\frac{a_g(B)}{\omega}\right)^{(n-2)/(n-1)}.
\]
\end{corollary}
\begin{proof}
Lemma~\ref{lem:g-enclosure} supplies a minimizing enclosure of full perimeter
$a_g(B)$. By Lemma~\ref{lem:g-detachment} its frontier is detached, locally
perimeter minimizing, and outer minimizing, and its closed end exterior is
connected and complete. The restricted metric keeps the scalar-curvature and
end hypotheses and is smooth through the frontier, so
Theorem~\ref{thm:intro-main} applies to this exterior and gives the displayed
bound.
\end{proof}
\section{The conformal flow}
\label{sec:rpi:flow-localization}
Bray's conformal flow changes the metric while keeping the horizon area fixed.
The idea is to push a positive conformal factor out from the horizon, at a rate
set by a harmonic ``velocity'' potential that vanishes on the horizon and tends
to a constant at infinity. Nonnegative scalar curvature is preserved
automatically because the conformal factor is harmonic. This section builds the
flow for $n\ge8$ and records that it preserves area; later sections use the
flow to decrease mass.
\subsection{The ambient manifold and the filled side}
Extend the original metric $g_0=h$ a short distance behind its compact frontier,
and truncate on that buried side at a smooth compact boundary $B$. Write
$(M,g_0)$ for the resulting smooth ambient exterior and $J$ for its filled
side; $J$ contains a fixed collar of $B$. The metric on the buried side need
not have nonnegative scalar curvature.
Use the canonical closed representative of $J$, whose frontier is the full
perimeter support. Its open complement is connected. There is one component
containing the coordinate end; any other component $U$ is bounded with
$\partial U$ in the original frontier, which has finite $(n-1)$-measure and a
singular part of zero measure. So $U$ has finite perimeter, and on regular
boundary charts its normal is opposite to the filled-side normal. Filling $U$
then gives $P_{g_0}(J\cup U)=P_{g_0}(J)-P_{g_0}(U)$, contradicting outer
minimization. Hence the open exterior is connected.
Local perimeter minimality of the original frontier, together with compactness,
gives a single radius for all sufficiently small local comparisons. We may
replace $J$ by its outermost minimizing enclosure $K_0$: outer minimization
preserves the area, and the lattice comparison preserves the local minimality.
Write $E_0=M\setminus K_0$. Nonnegative scalar curvature holds on $E_0$ and
hence on every later $E_t\subset E_0$. The buried boundary stays buried.
\subsection{The flow equations}
Set $k=n-1$ and $d=n-2$. The flow has metrics
$g_t=u_t^{4/d}g_0$ and filled regions $K_t$, with $E_t=M\setminus K_t$, given
by
\begin{equation}\label{eq:rpi:flow-equations}
u_t=1+\int_0^t v_s\,ds,\qquad g_t=u_t^{4/d}g_0,\qquad
\Delta_{g_0}v_t=0\ \text{in }E_t,\quad
v_t=0\ \text{on }K_t,\quad
v_t\longrightarrow-e^{-t}.
\end{equation}
Then $v_t=0$ and $u_t=1$ on the same fixed collar of $B$. Two harmonic
potentials appear and must be kept apart. The $g_0$-capacitary potential is
$1+e^t v_t$; the $g_t$-capacitary potential, used later, is
\begin{equation}\label{eq:rpi:capacity-potential}
w_t=1+\frac{v_t}{u_t}\quad\text{on }E_t,\qquad w_t=1\ \text{on }K_t.
\end{equation}
Both are constant on the buried collar, so neither feels a mean-curvature
condition at $B$. Capacity test functions are compactly supported relative to
$M$, smooth up to $B$, and equal to one near $K_t$; they need not vanish on $B$.
Every regular level below one avoids the buried collar, so a capacity--flux
calculation sees only the level boundary and the sphere at infinity.
The reference metric $g_0$ stays smooth. The working metric $g_t$ is smooth on
the open exterior, where $u_t$ is $g_0$-harmonic and
$R_{g_t}=u_t^{-4/d}R_{g_0}\ge0$. Across a singular frontier one uses comparison
and H\"older estimates, then the one-sided regularity and relative forcing of
Section~\ref{sec:rpi:regular-patches}. Once a new frontier is detached, it sits
in a smooth neighborhood, and the smoothing and mass--capacity arguments take
place there, away from the original frontier and from $B$.
\subsection{Construction and preserved area}
\label{sec:rpi:flow-construction}
\begin{lemma}[Construction and area of the conformal flow]
\label{lem:rpi:flow-construction}
Let $K_0$ be the initial outermost minimizing enclosure above, and put
$E_0=M\setminus K_0$, $\Sigma_0=\partial K_0$, $A_0=P_{g_0}(K_0)$, with
perimeter relative to $\operatorname{int}M$ and counting the whole frontier.
There are increasing closed filled sets $K_t\supset K_0$, with connected end
exteriors $E_t=M\setminus K_t$, and positive continuous functions $u_t$ solving
\eqref{eq:rpi:flow-equations}. For every $t$ we have $e^{-t}\le u_t\le1$, each
$K_t$ locally minimizes perimeter in the metric $g_t$ against variations in
balls below the initial comparison radius, and $K_t$ is the outermost
$g_t$-minimizing enclosure of $K_0$, so
\begin{equation}\label{eq:rpi:flow-area}
P_{g_t}(K_t)=A_0.
\end{equation}
On each finite time interval the conductors have common compact support, the
factors have a common positive lower bound and H\"older modulus, and the
frontiers obey uniform two-phase density and almost-minimizing estimates: there
are $\alpha_T>0$, $C_T<\infty$, $r_T>0$ such that
\begin{equation}\label{eq:rpi:flow-almost-minimizing}
P_{g_0}(K_t;B_r(x))\le P_{g_0}(L;B_r(x))+C_T r^{k+\alpha_T}
\end{equation}
whenever $0\le t\le T$, $r<r_T$, and $L\mathbin{\triangle}K_t\Subset B_r(x)$.
\end{lemma}
The discrete method is Bray's \cite{Bray2001}; its singular-frontier form is
\cite[Section~2.1]{BiZhu2026v2}. We follow the order of
\cite[Lemmas~2.21, 2.23 and Corollary~2.22]{BiZhu2026v2}: area preservation is
proved before the nesting of limiting outermost hulls. Freezing on a discrete
conductor does not by itself give freezing on a possibly larger limiting
conductor.
The proof has three stages, split over the next two subsections. First a static
weighted enclosure problem supplies compactness and boundary estimates for one
conductor. Then those estimates are applied to the discrete flow, and the area
loss is shown to tend to zero. Finally the minimum area of each limiting metric
is identified; that equality gives nesting and the integral evolution.
\subsection{A static weighted enclosure}
The discrete flow is built from a static problem: minimize a perimeter with a
positive continuous weight against variations, then read off a harmonic escape
potential. We isolate this because the same statement gives compactness, the
confinement radius, the two-phase density, and the H\"older modulus of the
velocity all at once.
Write $P_a(L;U)=\int_{U\cap\partial^*L}a\,d\mathcal H^k_{g_0}$ for the weighted
perimeter.
\begin{lemma}[Weighted enclosure and its potential]
\label{lem:rpi:static-hull-potential}
Let $a$ be continuous with $0<c\le a\le1$ and $a=1$ on a compact
finite-perimeter filled set $K_0$ containing a collar of $B$, which is locally
perimeter minimizing in $g_0$ with comparison radius $r_0$. Put
$A_0=P_{g_0}(K_0)$.
\begin{enumerate}
\item Every compact finite-perimeter obstacle has a unique outermost
$P_a$-minimizing enclosure, for any continuous positive weight bounded
above and below.
\item Every $P_a$-minimizing enclosure $K\supset K_0$ has $P_a(K)\le A_0$ and
lies in a ball $B_{R_*(c)}$ depending only on $c,A_0$, the initial
support, and the reference geometry. It locally minimizes $P_a$ against
unconstrained variations in balls of radius below $r_0$; its canonical
representative has a connected open end exterior and uniform two-phase
volume and perimeter density bounds.
\item There is a unique escape potential $q_K$, harmonic in the exterior, zero
on $K$, tending to one at infinity, continuous with a H\"older modulus
depending only on the same data, and $0<q_K<1$ in the exterior.
\item If $\operatorname{osc}_{B_r}a\le C_a r^\alpha$ for $0<\alpha<1$, then for
local competitors $L$,
\[
P_{g_0}(K;B_r)\le P_{g_0}(L;B_r)+Cr^{k+\alpha},
\]
with $C$ depending also on $C_a$.
\end{enumerate}
\end{lemma}
\begin{proof}
\emph{Confinement.} Choose $R_0$ containing $K_0$ and the compact core, and
$\gamma\ge1$ with $\gamma^{-2}\delta\le g_0\le\gamma^2\delta$ in the end. Put
$a_-=c\gamma^{-k}$, $a_+=\gamma^k$. For almost every $R>R_0$ the truncation by
$B_R$ is admissible, and cancellation of the common interior perimeter gives
\begin{equation}\label{eq:rpi:flow-truncation}
P_\delta(L;\{|x|>R\})\le\frac{a_+}{a_-}\mathcal H^k_\delta(L\cap S_R).
\end{equation}
Euclidean isoperimetry on the cut tail and the coarea formula give, with
$V(R)=|L\cap\{|x|>R\}|_\delta$,
\[
V(R)^{k/n}\le C_{\rm iso}\Bigl(1+\frac{a_+}{a_-}\Bigr)[-V'(R)].
\]
At a regular cutting radius $R_1\in[R_0,R_0+1]$, a second isoperimetric estimate
gives $V(R_1)\le V_*:=[C_{\rm iso}(a_-^{-1}A_0+\mathcal H^k_\delta(S_{R_0+1}))]^{n/k}$.
While $V>0$ we have
$\frac{d}{dR}V^{1/n}\le-[nC_{\rm iso}(1+a_+/a_-)]^{-1}$, so the tail vanishes by
\begin{equation}\label{eq:rpi:flow-hull-radius}
R_*(c)=R_0+1+nC_{\rm iso}\Bigl(1+\frac{a_+}{a_-}\Bigr)V_*^{1/n}.
\end{equation}
For the canonical representative zero tail volume means actual containment.
\emph{Existence and outermostness.} Minimize on each $B_Q$; BV compactness and
weighted lower semicontinuity give a constrained minimizer, and
\eqref{eq:rpi:flow-hull-radius} is independent of $Q$, so a subsequence
converges on one fixed support ball and the limit is a global minimizer.
Submodularity shows the union and intersection of two minimizers are again
minimizers; the one of maximal background volume contains all others. That is
the unique outermost hull.
\emph{Removal of the initial obstacle.} Let $K$ be any minimizing enclosure.
Initial local minimality of $K_0$ and global enclosure minimality of $K$, via
submodularity, give $P_a(K;B_r(x))\le P_a(L;B_r(x))$ for
$L\mathbin{\triangle}K\Subset B_r(x)$ with $r<r_0$. Thus $K$ is a local
background quasiminimizer with factor $c^{-1}$, including balls crossing $K_0$.
\emph{Two-phase density.} With $m(r)=\Vol(K\cap B_r(x))$ and
$e(r)=\Vol(B_r(x)\setminus K)$, deleting or filling $B_r(x)$ and using BV
slicing and quasiminimality give $P_{g_0}(K;B_r(x))\le Qm'(r)$ and
$\le Qe'(r)$ for almost every $r$. Relative isoperimetry then gives
$\min\{m(r),e(r)\}\ge[nI(Q+1)]^{-n}r^n$, and perimeter-measure monotonicity
extends $c_*r^k\le P_{g_0}(K;B_r(x))\le C_*r^k$ to every small radius. These
bounds identify $\partial K$ with $\operatorname{supp}|D\mathbf 1_K|$ at every
frontier point and give the connected end exterior by the filling argument.
\emph{The escape potential.} Let $\mathcal D^{1,2}(M)$ be the completion in the
gradient norm of smooth functions compactly supported relative to $M$. Minimize
$\int_M|\nabla W|^2$ over functions whose quasi-continuous representative is at
least one quasi-everywhere on $K$. The asymptotically flat Sobolev inequality
and the Hilbert-space method give a unique minimizer $W_K$ with $0\le W_K\le1$;
variations give that $q_K=1-W_K$ is locally $H^1$, zero-extended over $K$,
$0\le q_K\le1$, with
\begin{equation}\label{eq:rpi:flow-weak-potential}
\Delta_{g_0}q_K\ge0\ \text{weakly away from }B,\qquad
\Delta_{g_0}q_K=0\ \text{in }M\setminus K.
\end{equation}
A radial supersolution gives $q_K\to1$ at infinity.
\emph{Boundary contraction.} Fix $x\in\partial K$ and work in uniform
coordinates. The density bounds give $|K\cap B_r(x)|\ge d_*r^n$; choose
$\kappa$ small with $|B_{\kappa r}(x)|\le d_*r^n/2$. Coarea supplies
$\rho\in[\kappa r,r]$ with $\mathcal H^k_\delta(K\cap\partial B_\rho(x))\ge
d'_*r^k$. Let $H$ be the harmonic replacement in $B_\rho(x)$ with trace $q_K$;
weak subharmonicity and the maximum principle give $q_K\le H$, and with
$M(r)=\operatorname{ess\,sup}_{B_r(x)}q_K$, the exterior-indicator datum and the
uniform Poisson-kernel lower bound give
\[
H(y)\le M(r)\int_{\partial B_\rho(x)}P_\rho(y,\xi)\chi(\xi)\,dS_\delta(\xi),
\]
where $\chi=\mathbf1_{(M\setminus K)\cap\partial B_\rho(x)}$. The datum is the
exterior indicator, not $1-\chi$: only the former dominates the possibly
positive trace of $q_K$. The kernel's missing mass on the conductor portion is
at least a uniform $\delta_*>0$, so iterating from $q_K\le1$ gives
\[
M(\kappa r/2)\le(1-\delta_*)M(r),\qquad
\operatorname*{ess\,sup}_{B_r(x)}q_K\le C_*(r/r_*)^\alpha .
\]
Interior smoothness, the strong maximum principle on the connected exterior,
and an interior gradient estimate give $0<q_K<1$ and the full spatial H\"older
bound
\begin{equation}\label{eq:rpi:flow-holder-input}
\|q_K\|_{C^{0,\alpha}(M,g_0)}\le C_* .
\end{equation}
\emph{The scale-invariant comparison.} Suppose $\operatorname{osc}_{B_r}a\le
C_a r^\alpha$. If $P_{g_0}(L;B_r)\ge P_{g_0}(K;B_r)$ the claim is immediate;
otherwise both perimeters are at most $C_*r^k$, and the exact weighted
comparison gives
\[
P_{g_0}(K;B_r)\le\frac{\sup_{B_r}a}{\inf_{B_r}a}P_{g_0}(L;B_r)
\le P_{g_0}(L;B_r)+Cr^\alpha P_{g_0}(L;B_r)
\le P_{g_0}(L;B_r)+Cr^{k+\alpha}.
\]
This proves item (iv).
\end{proof}
\subsection{The discrete flow and its area loss}
We build the flow in discrete time. Fix $0<\epsilon<1/2$, set
$u_0^\epsilon=1$ and $K_0^\epsilon=K_0$, and write
$h_j^\epsilon=(u_j^\epsilon)^{4/d}g_0$, $a_j^\epsilon=(1-\epsilon)^j$. For
$j\ge1$ let $K_j^\epsilon$ be the outermost $h_j^\epsilon$-minimizing
enclosure of $K_{j-1}^\epsilon$. Let $v_j^\epsilon$ be the $g_0$-harmonic
function on $M\setminus K_j^\epsilon$ with value $0$ on $K_j^\epsilon$ and end
value $-a_j^\epsilon$, extended by zero over $K_j^\epsilon$. Define
\begin{equation}\label{eq:rpi:flow-euler}
u_{j+1}^\epsilon=u_j^\epsilon+\epsilon v_j^\epsilon,\quad
u_t^\epsilon=u_j^\epsilon+(t-j\epsilon)v_j^\epsilon,\quad
K_t^\epsilon=K_j^\epsilon,\quad
v_t^\epsilon=v_j^\epsilon
\end{equation}
on $j\epsilon\le t<(j+1)\epsilon$, using the new hull and velocity at a grid
endpoint. Then $u_t^\epsilon=1+\int_0^t v_s^\epsilon\,ds$ exactly. The maximum
principle gives $-a_j^\epsilon\le v_j^\epsilon\le0$ and
$(1-\epsilon)^{j+1}\le u_t^\epsilon\le1$; concavity of $\log(1-\theta)$
improves this to $u_t^\epsilon\ge(1-\epsilon)^{t/\epsilon}\ge4^{-t}$ for
$0\le t\le T$. Successive velocities are nondecreasing,
$v_j^\epsilon\le v_{j+1}^\epsilon\le0$, by comparison across nested conductors.
The key point is that each grid conductor is the outermost minimizing enclosure
of the fixed $K_0$ in its own metric, which lets the static lemma apply at every
step.
\begin{lemma}[Uniform estimates for the discrete flow]
\label{lem:rpi:discrete-flow}
For every finite $T$ the induction defines the flow on $[0,T]$. Every grid
conductor $K_j^\epsilon$ is the outermost $h_j^\epsilon$-minimizing enclosure of
the fixed $K_0$. The conductors are nested with common compact support. Their
velocities and the factors $u_t^\epsilon$ have a common spatial H\"older modulus;
$u_t^\epsilon$ is uniformly Lipschitz in time. The frontiers have uniform
density and background almost-minimizing estimates. The grid areas
$A_j^\epsilon=P_{h_j^\epsilon}(K_j^\epsilon)$ are nonincreasing, and the area
is frozen between grid times.
\end{lemma}
\begin{proof}
\emph{Fixed-obstacle minimality.} Suppose $K_{j-1}^\epsilon$ is the outermost
$h_{j-1}^\epsilon$-minimizing enclosure of $K_0$, and let $L\supset K_0$ be a
competitor. The two consecutive metrics agree on $K_{j-1}^\epsilon$, because
$v_{j-1}^\epsilon$ vanishes there, and $\partial(L\cap K_{j-1}^\epsilon)$ is
supported in that same set. Hence
\[
P_{h_j^\epsilon}(K_{j-1}^\epsilon)
=P_{h_{j-1}^\epsilon}(K_{j-1}^\epsilon)
\le P_{h_{j-1}^\epsilon}(L\cap K_{j-1}^\epsilon)
=P_{h_j^\epsilon}(L\cap K_{j-1}^\epsilon),
\]
and perimeter submodularity gives $P_{h_j^\epsilon}(L\cup K_{j-1}^\epsilon)\le
P_{h_j^\epsilon}(L)$. The left set is admissible for $K_j^\epsilon$, so
$K_j^\epsilon$ is globally minimizing; outermostness forces every minimizer
$L\subset K_j^\epsilon$. The induction starts at the actual outermost initial
hull.
\emph{Closing the induction.} At step $j$ the constructed factor defines the
weighted hull problem, and the minimality just proved lets
Lemma~\ref{lem:rpi:static-hull-potential} apply with $a=(u_j^\epsilon)^p$,
$p=2k/d$, and $c=4^{-pT}$. It gives the densities and the continuous potential
$v_j^\epsilon=-a_j^\epsilon q_{K_j^\epsilon}$, with
$\|v_j^\epsilon\|_{C^{0,\alpha}}\le C_T$. In particular $v_j^\epsilon$ vanishes
on the whole canonical conductor, so the next factor is continuous and agrees
with $u_j^\epsilon$ on $K_j^\epsilon$. This closes the induction.
The exact discrete integral identity gives the uniform spatial H\"older bound
for $u_t^\epsilon$, while $|v_j^\epsilon|\le1$ gives the time Lipschitz bound.
Hence $\operatorname{osc}_{B_r}(u_j^\epsilon)^p\le C_T r^\alpha$, and item (iv)
of the static lemma gives the almost-minimizing estimate with error
$C_T r^{k+\alpha}$.
\emph{Uniform support and area.} The confinement radius
\eqref{eq:rpi:flow-hull-radius} with the fixed obstacle $K_0$, area bound $A_0$,
and $c=4^{-pT}$ gives $K_t^\epsilon\subset B_{R_T}$ and
$\Vol_{g_0}(K_t^\epsilon)\le V_T$ on $[0,T]$. Set
$A_j^\epsilon=P_{h_j^\epsilon}(K_j^\epsilon)$. Freezing on the previous
conductor and its admissibility at the next step give
$0\le A_j^\epsilon\le A_{j-1}^\epsilon\le A_0$. The metric at an intermediate
time need not make the frozen conductor globally minimizing, but its area is
$A_j^\epsilon$ because $v_j^\epsilon=0$ on that frontier.
\end{proof}
\begin{proposition}[Vanishing discrete area loss]
\label{prop:rpi:discrete-area}
For every finite $T$, the areas of the discrete frontiers tend uniformly to
$A_0$ for $0\le t\le T$ as $\epsilon\downarrow0$.
\end{proposition}
\begin{proof}
Global minimality in the previous grid metric, with $K_j^\epsilon$ as
competitor, and Bernoulli's inequality with $v_{j-1}^\epsilon\le0$ give
\[
0\le A_{j-1}^\epsilon-A_j^\epsilon
\le-p\epsilon\int_{\partial^*K_j^\epsilon}
\frac{v_{j-1}^\epsilon}{u_{j-1}^\epsilon}
\,d\mathcal H^k_{h_{j-1}^\epsilon}
\le C_T\epsilon A_0\sup_{\partial K_j^\epsilon}|v_{j-1}^\epsilon|.
\]
Separate the transitions where $\Vol_{g_0}(K_j^\epsilon\setminus
K_{j-1}^\epsilon)>V_T\sqrt\epsilon$ from the rest. Nesting and the support bound
make the first kind number at most $\epsilon^{-1/2}$, so their total loss is
$O_T(\sqrt\epsilon)$. For a transition of the second kind the two-phase density
gives the frontiers within Hausdorff distance
$C_T(V_T\sqrt\epsilon)^{1/n}$; the old velocity vanishes on the old frontier, so
its H\"older modulus bounds its supremum on the new frontier by
$C_T\epsilon^{\alpha/(2n)}$. Summing the $\epsilon$-weighted losses over at most
$T/\epsilon+1$ transitions gives
\[
0\le A_0-A_j^\epsilon\le C_T(\sqrt\epsilon+\epsilon^{\alpha/(2n)})
\longrightarrow0
\]
uniformly for $j\epsilon\le T$; intermediate times follow since the current
frontier has frozen area.
\end{proof}
\subsection{The limiting flow}
We now pass the discrete flow to a limit and prove
Lemma~\ref{lem:rpi:flow-construction}. The order is: compactness of the
factors, identification of the minimum enclosure area, then nesting and the
integral evolution.
\emph{Compactness of the factors.} The factors $u_t^\epsilon$ are uniformly
Lipschitz in time and H\"older in space, so a mesh sequence tending to zero has
a locally uniformly convergent subsequence, jointly in space and time, to a
positive $u_t$; put $g_t=u_t^{4/d}g_0$. To keep the normalization at the end,
take $\beta=d/2$ and enlarge $R_T$ until $\Delta_{g_0}r^{-\beta}<0$ for
$r\ge R_T$. The function $1+v_j^\epsilon/a_j^\epsilon$ is harmonic there, lies
between zero and one on the inner sphere, and tends to zero at infinity, so
\[
0\le v_j^\epsilon+a_j^\epsilon\le a_j^\epsilon(R_T/r)^\beta .
\]
Integrating along the current step, with $a_\epsilon(t)=a_j^\epsilon(1-t+j\epsilon)$,
gives $|u_t^\epsilon-a_\epsilon(t)|\le T(R_T/r)^\beta$ on $0\le t\le T$,
$r\ge R_T$. Since $a_\epsilon(t)\to e^{-t}$ uniformly on $[0,T]$, local
uniform convergence upgrades to uniform convergence on $[0,T]\times M$, and the
constant velocity limit $-e^{-t}$ is retained.
\emph{The minimum enclosure area.} Fix $t$ and a bounded finite-perimeter
enclosure $L\supset K_0$. Grid minimality at $j=\lfloor t/\epsilon\rfloor$,
uniform convergence of the lagged metric, and the area-loss bound imply
\begin{equation}\label{eq:rpi:flow-competitor-lower-bound}
P_{g_t}(L)\ge A_0.
\end{equation}
All discrete velocities vanish on $K_0$, so $u_t=1$ there and $P_{g_t}(K_0)=A_0$.
The minimum enclosure perimeter at time $t$ is therefore exactly $A_0$.
Common support and the background perimeter bound give a subsequential BV limit
$\widehat K_t$ of the grid conductors, and lower semicontinuity in the
uniformly convergent weights gives $P_{g_t}(\widehat K_t)\le A_0$; by
\eqref{eq:rpi:flow-competitor-lower-bound} it is a minimizer. This also proves
perimeter convergence, and the density bounds upgrade it to Hausdorff
convergence of the frontiers. Define $K_t$ to be the outermost
$g_t$-minimizing enclosure of $K_0$; it has perimeter $A_0$, contains every
$\widehat K_t$, and inherits local weighted minimality plus the uniform
density and almost-minimizing estimates from the static lemma.
\emph{Nesting.} For $s>t$, monotonicity of the discrete factors gives
$u_s\le u_t$, and the minimum areas give
\[
A_0\le P_{g_s}(K_t)\le P_{g_t}(K_t)=A_0.
\]
Hence $u_s=u_t$ almost everywhere on $\partial K_t$, and continuity plus
frontier density improve this to
\begin{equation}\label{eq:rpi:flow-frontier-freezing}
u_s=u_t\quad\text{on }\partial K_t.
\end{equation}
Take a joint subsequential limit $(\widehat K_s,\widehat v_s)$ of the grid
conductors and velocities at time $s$. The boundary H\"older estimate, interior
harmonic compactness, and the tail comparison identify $\widehat v_s$ as the
harmonic function zero on $\widehat K_s$ with end value $-e^{-s}$, strictly
negative on its connected exterior. The discrete integral identity and velocity
monotonicity give $u_t^\epsilon-u_s^\epsilon\ge(s-t)(-v_s^\epsilon)$, and
passing to the subsequence,
\[
u_t-u_s\ge(s-t)(-\widehat v_s)>0\quad\text{on }M\setminus\widehat K_s.
\]
Together with \eqref{eq:rpi:flow-frontier-freezing} this forces
$\partial K_t\subset\widehat K_s$, and the exterior cannot cross
$\partial K_t$, so
\begin{equation}\label{eq:rpi:flow-limit-squeeze}
K_t\subset\widehat K_s\subset K_s\qquad(t<s).
\end{equation}
\emph{The integral evolution.} The volume $V(t)=\Vol_{g_0}(K_t)$ is bounded and
nondecreasing, so its positive left jumps are countable. At a nonjump time $t$
the squeeze \eqref{eq:rpi:flow-limit-squeeze}, taken over rational $r<t$, gives
$\overline{\bigcup_{r<t}K_r}=K_t$, and the canonical identity
$K_t=\operatorname{supp}(\chi_{K_t}\,dV_{g_0})$ makes $\widehat K_t=K_t$. Thus
all subsequential grid limits agree, and compactness makes the entire chosen
sequence converge in $L^1$ and in Hausdorff distance of frontiers. The boundary
H\"older estimates, interior compactness, uniqueness, and uniform tail
comparison give $v_t^\epsilon\to v_t$ in $C^0$, where $v_t$ is the harmonic
function on $E_t$ zero on $K_t$ with end value $-e^{-t}$. Dominated convergence
in the exact discrete identity gives $u_t=1+\int_0^t v_s\,ds$.
Now nesting and the integral identity give $u_s=u_t$ throughout $K_t$ for
$s\ge t$, since each later velocity vanishes there. The end normalization
$v_s\ge-e^{-s}$ improves the lower bound to $u_t\ge e^{-t}$. It also shows that
$u_t$ is $g_0$-harmonic in $E_t$, since each earlier velocity is harmonic there
and interior estimates justify integration on compact subsets. Because
$v_t=0$ and $u_t=1$ on the buried collar, and
\[
\Delta_{g_t}\Bigl(\frac{v_t}{u_t}\Bigr)
=u_t^{-(n+2)/d}\Bigl(\Delta_{g_0}v_t-\frac{v_t}{u_t}\Delta_{g_0}u_t\Bigr)=0
\quad\text{in }E_t,
\]
the function $w_t=1+v_t/u_t$ is the $g_t$-harmonic capacitary potential, equal
to one on the frontier and zero at infinity, with $0\le w_t\le1$. A diagonal
choice of convergent meshes over $T=1,2,\ldots$ completes the construction for
all $t\ge0$. The area identity \eqref{eq:rpi:flow-area} holds at every time,
including jump times, and the local weighted comparison with its H\"older-weight
conversion gives \eqref{eq:rpi:flow-almost-minimizing}.
\section{Regular patches and separation of frontiers}
\label{sec:rpi:regular-patches}
To run the mass--capacity argument we need distinct flow frontiers to be
disjoint, including at singular points. On regular patches the separation of two
nested frontiers is described by a scalar graph equation. The difficulty is
that a strict Hopf derivative at each fixed scale does not survive
normalization: the harmonic forcing shrinks along with the graphs. So we need a
relative estimate, one that compares the forcing to the size of the harmonic
function itself.
Fix $T<\infty$, and write $k=n-1$ and $d=n-2$. We use the flow already built:
$K_t$ is the closed filled side, $E_t=M\setminus K_t$ the connected end
exterior, $\Sigma_t=\operatorname{supp}|D\chi_{K_t}|$ the frontier, with the
normal pointing from the filled side into the exterior. In $g_0$ the frontiers
have uniform local perimeter and two-phase density bounds and an
almost-minimizing error $\beta r^{k+\mu}$, $\mu>0$. Recall
$g_t=u_t^{4/d}g_0$ with $0<c_T\le u_t\le1$, $|v_t|\le1$, $u_t=1+\int_0^t v_s\,
ds$, and each $v_t$ harmonic in $E_t$ and zero on $K_t$; the factors share a
background H\"older modulus.
\begin{lemma}[Simultaneous regular neighborhoods]
\label{lem:rpi:simultaneous-regularity}
Fix a regular point $p$ of one $\Sigma_t$ and a coordinate plane parallel to its
tangent plane there. Given a sufficiently small slope bound, there are nested
coordinate neighborhoods $U\Subset V$ of $p$ such that every frontier meeting
$U$ is a single graph in $V$ over that plane, with the prescribed slope bound
and a uniform $C^{1,\theta}$ estimate. On smaller neighborhoods, every fixed
finite order of one-sided regularity of the graphs, harmonic velocities, and
conformal factors is uniform for $0\le t\le T$. The same conclusions hold
uniformly for the rescaled time families on each fixed compact regular patch of
a limiting cone.
\end{lemma}
\begin{proof}
\emph{A common graphical neighborhood.} A power-almost-minimizing sequence has
a subsequence converging in $L^1$ with local perimeter convergence; if the limit
has a regular point, the boundaries converge as single $C^{1,\theta}$ graphs on
a smaller neighborhood. Applying this to frontiers meeting shrinking balls at
$p$: the filled sets lie on the same ordered side, so every tangent filled set
of the limit is ordered with the tangent half-space of the reference sheet. Its
minimizing boundary cone then lies in a closed half-space. On the spherical link
the nonnegative height $z_\perp$ satisfies $\Delta_{\rm link}z_\perp=-(k-1)
z_\perp$, and integration forces $z_\perp=0$; with multiplicity one the cone is
a plane. Density-one regularity makes the limit a single graph, giving fixed
$U\Subset V$ for the whole family. If uniform neighborhoods failed, a sequence
of radii $r_j\downarrow0$ with failing points would contradict this fixed
graphical convergence after shrinking the cylinders.
\emph{One-sided derivatives and bootstrap.} Within the common graphical region,
flatten the boundary and reflect the negative exterior function $v_\sigma$ oddly
across it. The zero trace makes tangential derivatives vanish and odd reflection
matches the normal derivative, giving a uniformly controlled $C^{1,\alpha}$
extension, nonnegative on the interior side, dominating the original. Integrating
these extensions gives $\widetilde u_t\ge u_t$ on the interior with
$\widetilde u_t=u_t$ on $\overline{E_t}$ and $\|\widetilde u_t\|_{C^{1,\alpha}}
\le C$. The metric $\widetilde g_t=\widetilde u_t^{4/d}g_0$ is a uniformly
controlled $C^{1,\alpha}$ extension. Since $K_t$ is locally minimizing in $g_t$
and the two area functionals agree on $\Sigma_t$, it is locally minimizing in
$\widetilde g_t$; the minimal graph equation then improves the graphs to
$C^{2,\alpha}$. Feeding these improved domains back into the harmonic velocities
and iterating on nested neighborhoods gives every finite order. At higher stages
the extensions need only match one-sided boundary jets, since stationarity is
already established; no uniform Hopf lower bound is needed. All estimates are
uniform in the rescaling index because the rescaled smooth backgrounds are
uniformly bounded.
\end{proof}
The next estimate is relative: it controls how a normal derivative changes when
transferred between nearby graphs, divided by the size of the harmonic function.
\begin{lemma}[Relative harmonic derivatives]
\label{lem:rpi:relative-harmonic}
Consider uniformly $C^{2,\alpha}$ graphical domains in a fixed cylinder with
uniformly elliptic $C^{2,\alpha}$ metrics. Let $z_i>0$ be harmonic on the
exterior side with zero boundary trace, and let $A_i$ be interior points at
distance at least a fixed fraction of the cylinder radius from graph and lateral
boundary. Put $a_i=z_i(A_i)$. Then on nested central cylinders
\[
\|z_i\|_{C^{2,\alpha}}\le Ca_i,\qquad
ca_i\le\partial_\nu z_i\le Ca_i
\]
on the boundary, with $\nu$ pointing into the exterior. If a second graph lies
in the closure of the exterior and converges to the first in $C^1$, then on the
second graph the derivative in its corresponding exterior normal also lies
between $ca_i$ and $Ca_i$ for large $i$. All constants are relative to $a_i$;
no positive lower bound for $a_i$ is needed.
\end{lemma}
\begin{proof}
Normalize by $a_i$. Interior Harnack chains and the boundary Carleson estimate
bound the normalized function on a smaller cylinder, and boundary Schauder
estimates give the $C^{2,\alpha}$ bound. A fixed interior tangent ball supplies a
comparison value, and the tangent-ball barrier gives the two-sided derivative
bound. Joining corresponding points of the two graphs by the common graph
coordinate segment, the Hessian bound and convergence of the normals show that
the error in transferring the derivative is at most
$Ca_i(\text{height difference}+\text{normal difference})$, which is below half
the lower bound for large $i$. The factor $a_i$ multiplies both bound and error,
even as $a_i\to0$.
\end{proof}
\emph{Two flow times.} Fix $0\le s<t\le T$ and suppose a simultaneous blow-up at
a contact point has a common regular cone patch. Nesting traps every
intermediate filled set between the endpoints, and uniform almost-minimizing
compactness gives single-sheet $C^{1,\theta}$ graphs for all intermediate times,
converging in $C^1$ on a smaller patch. The relevant harmonic functions are
\begin{equation}\label{eq:rpi:relative-integrands}
z_\sigma=-\frac{v_\sigma}{u_s}\quad\text{on }E_\sigma,\qquad s\le\sigma\le t,
\end{equation}
positive and $g_s$-harmonic on the exterior with zero trace on the frontier; the
denominator is fixed at time $s$. Choose a common corkscrew point $A_i$ above all
intermediate graphs and set $a_{i,\sigma}=z_\sigma(A_i)>0$. The lemma transfers
each normal derivative to the later graph with relative error $o(1)a_{i,\sigma}$.
With $\lambda_i=\int_s^t a_{i,\sigma}\,d\sigma>0$ and the exact identity
\begin{equation}\label{eq:rpi:relative-integral}
1-\frac{u_t}{u_s}=\int_s^t z_\sigma\,d\sigma,
\end{equation}
dominated convergence in exterior difference quotients gives
\begin{equation}\label{eq:rpi:relative-forcing}
-C\lambda_i\le\partial_{\nu_t}^{g_s}\Bigl(\frac{u_t}{u_s}\Bigr)\le-c\lambda_i.
\end{equation}
The ratio $u_t/u_s$ is uniformly positive, so the conformal mean-curvature
factor preserves these bounds. Only exterior traces are differentiated.
\subsection{The graph equation and regular contact}
Both the flow and the prescribed-curvature enclosure use the same scalar graph
equation for two nested frontiers, the earlier one measured in a metric $g$ that
we may normalize at the contact point to be Euclidean. Write $g_i$ for its
rescaling at radius $r_i$, and use $g_i$-unit normals and $g_i$-normal
exponential graphs.
\emph{The equation.} The linearization of $H=\divg\nu$ at the earlier minimal
graph is $-J_i$, where
\[
J_i=\Delta_{\Sigma_i,g_i}+|A_i|_{g_i}^2+\Ric_{g_i}(\nu_i,\nu_i).
\]
The nonlinear remainder of the mean-curvature graph operator writes as
$Q_i\zeta_i=Q_i^{ab}\partial_{ab}\zeta_i+Q_i^a\partial_a\zeta_i+Q_i^0\zeta_i$
with coefficients tending locally to zero. Let the later metric be
$g'_i=U_i^{4/d}g_i$ with prescribed end-oriented curvature
$f_i(y)=r_i f(\eta_i(y))$, $\eta_i(y)=p+r_i y$. The conformal formula gives the
exact equation
\begin{equation}\label{eq:rpi:separation-graph-equation}
(J_i-Q_i)\zeta_i=F_i-U_i^{2/d}f_i,\qquad
F_i=\frac{2k}{d}U_i^{-1}\partial_{\nu'_i}^{g_i}U_i .
\end{equation}
For flow slices $f=0$ and $U_i$ is the rescaled ratio $u_t/u_s$;
\eqref{eq:rpi:relative-forcing} bounds $F_i$ from both sides.
\emph{Regular contact is impossible.} Fix $0\le s<t\le T$ and suppose
$p\in\Sigma_s\cap\Sigma_t$ is regular for at least one frontier. The
simultaneous regular-neighborhood lemma makes both graphs regular there, and
nesting puts $p$ on every intermediate frontier. Their velocities vanish at
$p$, so $u_t(p)=u_s(p)$. For each $\sigma\in[s,t]$, $z_\sigma=-v_\sigma/u_s$ is
positive and $g_s$-harmonic on its exterior, vanishes at $p$, and has a strictly
positive derivative in the common normal direction. The relative estimates,
applied to $q=u_t/u_s$, give
\[
\partial_\nu q(p)=-\int_s^t\partial_\nu z_\sigma(p)\,d\sigma<0 .
\]
Extend the later metric from its exterior as a $C^1$ metric. Its first jet at $p$
is determined one-sidedly, and the earlier metric has the matching jet. The
conformal mean-curvature formula gives
\[
H_{\Sigma_s}^{g_t}(p)
=q(p)^{-2/d}\Bigl(H_{\Sigma_s}^{g_s}(p)+\frac{2k}{d}q(p)^{-1}\partial_\nu q(p)\Bigr)<0,
\]
since the earlier curvature is zero (including $s=0$). The later graph has zero
curvature in the same extension metric. Because $K_s\subset K_t$, ordered
tangent-graph comparison gives $H_{\Sigma_t}^{g_t}(p)\le H_{\Sigma_s}^{g_t}(p)<0$,
a contradiction. So regular contact is impossible. Only common exterior first
jets were used; the ratio need not be constant on the later frontier.
\subsection{Singular separation from a single distance median}
\label{sec:rpi:median-separation}
We must exclude contact at a singular point. The method rescales a possible
contact and divides every graph separation by one number, the median distance
from the earlier support to the later one. This yields a single positive
supersolution on the regular part of the common tangent cone, whose infimum
still decays to zero near the vertex. A minimizing-boundary mean-value inequality
then rules it out. One median is chosen once for the whole ball, not patch by
patch; a patchwise normalization would let different patches carry different
constants.
\begin{proposition}[Separation of ordered supports]
\label{prop:rpi:singular-separation}
Let $K\subset K'$ be canonical closed filled sets in a smooth $n$-manifold,
$n\ge8$, with multiplicity-one perimeter supports $\Sigma,\Sigma'$. Put
$k=n-1$. On a neighbourhood of their possible contact set let $g_{\rm ref}$ be
smooth, and assume $P_{g_{\rm ref}}(E;B_r(x))\le P_{g_{\rm ref}}(L;B_r(x))+
\Lambda r^{k+\alpha}$ for $E=K,K'$, local competitors, and $0<r<r_0$. Assume
every intersection point is singular for both supports. Let
$g=a^{4/(n-2)}g_{\rm ref}$ with $a>0$ continuous be the metric measuring graph
heights. At a possible contact $p$, choose coordinates with $g(p)=\delta$ and
rescale by $\eta_i(y)=p+r_i y$, $r_i\downarrow0$. The power comparison gives
simultaneous minimizing-cone subsequences whose two limits coincide. For every
such subsequence with common cone $C$, on each compact regular patch (after
shrinking a larger patch) assume:
\begin{enumerate}
\item the rescaled supports converge as single smooth graphs, and the extensions
$g_i=r_i^{-2}\eta_i^*g\to\delta$ in $C^2$ with uniform local bounds;
\item the later support is the $g_i$-normal graph of $\zeta_i>0$ over the earlier
support, and
\begin{equation}\label{eq:rpi:separation-interface}
L_i\zeta_i=F_i\le0,\qquad L_i\longrightarrow J_C:=\Delta_C+|A_C|^2,
\end{equation}
with uniformly elliptic, uniformly $C^{0,\theta}$ operators converging
locally to that of $J_C$;
\item there are $\lambda_i>0$ and $c,C>0$, independent of $i$, with
$-C\lambda_i\le F_i\le0$ and $F_i\le-c\lambda_i$ on an inner ball.
\end{enumerate}
Then $\Sigma\cap\Sigma'=\varnothing$.
\end{proposition}
\begin{proof}
Suppose the supports meet at the origin.
\emph{A common cone with connected regular part.} The power comparison gives
strict perimeter compactness, so every simultaneous blow-up has a subsequence
converging to multiplicity-one minimizing boundary cones for $g_{\rm ref}(p)$,
nested and hence coincident; they are also stationary in the normalized
Euclidean metric. Their supports are connected (radial segments meet the
vertex), their singular sets have dimension at most $k-7$, and the same-metric
strong maximum principle \cite{Wickramasekera2014Maximum} identifies the two
supports since both contain the vertex; multiplicity one identifies the
measures. The regular part of the common cone $C$ is connected: restricting the
varifold to a regular component, stationarity extends across the singular set
(which has zero $\mathcal H^{k-1}$-measure) and each component is dilation
invariant with the vertex in its closure; two component closures meeting only
in the singular set would contradict the same strong maximum principle.
\emph{A median chosen first.} Let $D(y)=\dist_{\R^n}(y,\Sigma')$ and let $\mu_r$
be the perimeter measure of $\Sigma\cap B_r$. Define
\begin{equation}\label{eq:rpi:separation-median}
q(r)=\inf\Bigl\{a\ge0:\ \mu_r(\{D\le a\})\ge\tfrac12\mu_r(B_r)\Bigr\}.
\end{equation}
Atoms are allowed, so both $\{D\le q(r)\}$ and $\{D\ge q(r)\}$ carry at least
half the measure. The contact set has zero perimeter measure, so $q(r)>0$ for
small $r$, and $q(r)\le r$. Moreover $q(r)/r\to0$: otherwise a minimizing-cone
subsequence would make the rescaled supports coincide, contradicting the lower
bound on the rescaled median. With $f(r)=q(r)/r$ and any $R_i\downarrow0$, choose
$0<r_i\le R_i$ with $f(r_i)\ge\tfrac12\sup_{0<s\le R_i}f(s)$; then for
$0<\rho\le1$,
\begin{equation}\label{eq:rpi:median-record}
\frac{q(\rho r_i)}{q(r_i)}\le2\rho .
\end{equation}
Set $a_i=q(r_i)/r_i>0$, $\Sigma_i=r_i^{-1}\Sigma$, $\Sigma'_i=r_i^{-1}\Sigma'$.
This one number $a_i$ divides every graph separation on every patch.
\emph{Ambient distance is the normal height.} On a compact regular core of $C$,
for large $i$ the later support is the $g_i$-normal graph of $\zeta_i>0$ over
the earlier one with vanishing relative slopes, and the Lipschitz-graph estimate
gives
\begin{equation}\label{eq:rpi:distance-height}
\dist_{\R^n}(y,\Sigma'_i)=(1+o(1))\zeta_i(y)
\end{equation}
uniformly on a slightly smaller core, even when $\zeta_i$ is far below the
blow-up error.
\emph{One normalization on every patch.} Put $W_i=\zeta_i/a_i$. The lower
median set and \eqref{eq:rpi:distance-height} give a point with $W_i\le2$ in one
of finitely many regular patches. Weak Harnack for the nonnegative
supersolution $L_iW_i\le0$ (replacing the zero-order coefficient by its negative
part) gives a uniform $L^p$ bound, and connectedness of $\operatorname{Reg}C$
propagates it by finite chains to every fixed compact regular set:
\begin{equation}\label{eq:rpi:median-local-lp}
\|W_i\|_{L^p(K)}\le C_K\quad\text{for every }K\Subset\operatorname{Reg}C.
\end{equation}
\emph{Forcing before any limit.} In a ball where $F_i\le-c\lambda_i$, solve
$\widehat L_i\phi_i=-1$ with $\phi_i|_{\partial B}=0$; ellipticity and the
maximum principle give $\phi_i\ge c_0>0$ on a smaller ball, and comparison gives
$\zeta_i\ge c\lambda_i\phi_i$. Combined with \eqref{eq:rpi:median-local-lp} this
gives $\lambda_i/a_i\le C_K$, hence
\begin{equation}\label{eq:rpi:median-forcing-bound}
\|F_i/a_i\|_{L^\infty(K)}\le C_K .
\end{equation}
Bounded-right-side estimates, interpolated with the $L^p$ bounds and iterated on
nested radii, give uniform local $W^{2,q}$ bounds. One diagonal subsequence gives
a single function $W$ with local $C^1$ convergence on all of
$\operatorname{Reg}C$ and
\begin{equation}\label{eq:rpi:median-limit}
W\ge0,\quad W\in W^{2,q}_{\rm loc}(\operatorname{Reg}C),\quad
(\Delta_C+|A_C|^2)W\le0 .
\end{equation}
The upper median set gives points with $W_i\ge1/2$, so $W\not\equiv0$; since
$\Delta_CW\le-|A_C|^2W\le0$, the strong maximum principle and connectedness make
$W>0$ throughout the regular part.
\emph{The infimum near the vertex.} The record scales give, for every
$0<\rho\le1$, a point of a compact regular core inside $C\cap B_\rho$ with
$W_i\le3\rho$, so the same $W$ satisfies
\begin{equation}\label{eq:rpi:median-small-infimum}
\inf_{\operatorname{Reg}C\cap B_\rho}W\le3\rho .
\end{equation}
\emph{A bounded supersolution across the singular set.} Discarding the nonnegative
Jacobi term leaves $\Delta_CW\le0$; set $h=\min\{W,1\}$. Using the nonnegative
test $\eta^2(1-h)$, $h$ is a bounded positive weak Laplace supersolution on the
regular part with
\begin{equation}\label{eq:rpi:truncation-energy}
\int\eta^2|\nabla_Ch|^2\le4\int|\nabla_C\eta|^2 .
\end{equation}
The singular set has zero tangential two-capacity on compact regions: cover it by
balls $B_{\rho_j}(x_j)$ with $\sum_j\rho_j^{k-2}\to0$ (possible because its
dimension is at most $k-7$) and cutoffs $\theta_j$, and let $\chi=\min_j\theta_j$.
Then
\begin{equation}\label{eq:rpi:cone-capacity}
\int_C|\nabla_C\chi|^2\le C\sum_j\rho_j^{k-2}\longrightarrow0 .
\end{equation}
Using $\eta=\psi\chi$ in \eqref{eq:rpi:truncation-energy}, Fatou gives locally
finite energy for $h$, and $\psi h\chi\to\psi h$ in the local surface $W^{1,2}$
norm. So $h$ extends as a local Sobolev function on the whole minimizing
boundary and is a positive weak Laplace supersolution there.
\emph{The mean-value contradiction.} Apply the Bombieri--Giusti weak Harnack
inequality for minimizing boundaries \cite{BombieriGiusti1972Harnack}: for an
area-minimizing hypersurface in $B_R$ and a positive weak Laplace supersolution,
a normalized $L^p$ mean on $B_r$ is bounded by the infimum on $B_r$ when
$r\le\beta R$ and $0<p<k/(k-2)$. Applied to $h+\epsilon$ with $p=1$, the right
side is the essential infimum, which is zero by
\eqref{eq:rpi:median-small-infimum}; letting $\epsilon\downarrow0$ makes a
positive integral zero, a contradiction. So singular contact is impossible too.
\end{proof}
\subsection{The prescribed-curvature enclosure}
\label{sec:rpi:perturbed-candidate}
The mass--capacity comparison needs a smooth enclosing frontier with a strict
mean-curvature sign. We build it by adding a small positive volume forcing to a
harmonic conformal perturbation of the current metric; the resulting minimizer
has prescribed end-oriented curvature $-\delta h_0\le0$ and can be shown to
leave the original frontier.
Let $K$ be a compact canonical closed filled set with nonempty frontier $\Sigma$
and connected end exterior $E$. Work in either of two settings: (1) $K=K_s$ is a
flow conductor and $h=b^2g_s$ for a fixed constant $b>0$; (2) $h$ is smooth
through $\Sigma$ and on $E$, $K$ is locally perimeter minimizing in $h$ near
$\Sigma$, and the exterior has one coordinate end with
$h_{ab}=a_0^2\delta_{ab}+O_2(r^{-\tau_0})$. In both, $K$ is locally minimizing
in its continuous metric $h$. No outermost property is needed.
\emph{The harmonic perturbation.} Let $u$ be positive $h$-harmonic on $E$,
continuous up to $\Sigma$, with $u=1$ on $\Sigma$, $u<1$ in $E$, and
$u\to\ell\in(0,1)$ at infinity. Extend $u$ by one on $K$ and set
$h'=u^{4/d}h$; assume the end has $h'_{ab}=a^2\delta_{ab}+O_2(r^{-\tau})$ with
$a,\tau>0$, which will confine the candidate. The extension is H\"older: in the
flow case $z=u_s(1-u)$ is a bounded positive $g_0$-harmonic function equal to
$c\,q_{K_s}$, so bounded Dirichlet uniqueness gives
\[
u=1-\frac{c\,q_{K_s}}{u_s},
\]
which extends H\"older across $\Sigma$ by the H\"older escape potential; in the
smooth case $1-u=(1-\ell)q_K$ similarly. So $h'$ is a positive H\"older metric in
both cases.
Choose a smooth ambient $h_0$ that is zero on $K$, positive on $E$, integrable in
$h'$, and satisfies $|h_0(x)|\le C\dist(x,\Sigma)$ on compact sets. For
$\delta>0$ let $K'$ minimize
\[
\mathcal F(L)=P_{h'}(L)+\delta\int_{L\setminus K}h_0\,dV_{h'}
\]
among compact finite-perimeter supersets of $K$. A normal variation $\varphi\nu$
gives first variation $\int_{\partial K'}(H_\nu+\delta h_0)\varphi\,dA$, so the
prescribed end-oriented curvature is
\begin{equation}\label{eq:rpi:first-variation}
H_\nu=-\delta h_0\le0 .
\end{equation}
\emph{Existence, confinement, and a filled enclosure.} Choose $R_*$ enclosing
$K$ so that every coordinate sphere beyond $R_*$ has strictly positive outward
mean curvature in $h'$. On a truncation $B_Q$, $Q>R_*$, BV compactness and lower
semicontinuity give a minimizer; clipping a tail by a coordinate sphere decreases
perimeter and the nonnegative volume term, so the minimizer is supported in
$B_{R_*}$ independently of $Q$, and is global. Choose $\delta$ small enough that
$\delta\|h_0\|_{L^\infty}C_{\rm iso}\Vol_{h'}(B_{R_*})^{1/n}<1$. Then a bounded
complementary component $U$ satisfies
$\mathcal F(L\cup U)-\mathcal F(L)=-P_{h'}(U)+\delta\int_Uh_0\,dV_{h'}<0$ unless
$U$ has zero volume, so $K'$ is filled with a connected end exterior.
\emph{Local variations and graph estimates.} For a local competitor supported in
a small ball, $h'=h$ on $K$, the local minimality of $K$ and submodularity show
$K'$ is locally unconstrained minimizing, hence a quasi-minimizer; the comparison
gives the uniform almost-minimizing estimate. At a hypothetical cone patch,
small-excess regularity gives uniform $C^{1,\theta}$ candidate graphs; the
prescribed-curvature graph equation then improves them to every finite order, all
before any separation is assumed. Extending $1-u$ oddly through the original
graph and multiplying factor into a regular extension metric preserves the local
minimizer and its Euler equation $H_\nu^{h'}=-\delta h_0$.
\emph{Regular contact is forbidden.} Suppose the candidate touches the original
frontier regularly. There $h_0=0$, and the strict harmonic derivative makes the
original graph have strictly negative end-oriented curvature in $h'$, while the
candidate is minimal there. Ordered graph comparison contradicts this. Singular
contact is excluded as follows.
\emph{The candidate graph equation.} Apply
\eqref{eq:rpi:separation-graph-equation} with earlier metric $g=h$, later metric
$h'=u^{4/d}h$, and prescribed curvature $f=-\delta h_0$. Since $f$ is smooth and
zero on $K$, $|f(x)|\le C\dist(x,\Sigma)$, and at rescaled graph height $\zeta_i$
the distance to $\Sigma$ is at most $Cr_i\zeta_i$, so
$|U_i^{2/d}f_i|\le Cr_i^2\zeta_i$. Absorb this into the operator:
\[
L_i=J_i-Q_i+b_i,\qquad b_i=\frac{U_i^{2/d}f_i}{\zeta_i},\quad |b_i|\le Cr_i^2.
\]
The coefficient $b_i$ has a uniform H\"older bound because $f$ vanishes on the
earlier graph, expressing $f_i/\zeta_i$ as $r_i^2$ times an average of first
derivatives of $f$. Thus $L_i\to J_C$ with the coefficient control required by
Proposition~\ref{prop:rpi:singular-separation}. The strictly negative right side
$F_i\le0$ comes from the harmonic conformal factor: applying
Lemma~\ref{lem:rpi:relative-harmonic} to $1-u>0$ gives the patchwise two-sided
relative bounds for $F_i$. The sign of $f$ is irrelevant here; all parameters are
fixed during the argument.
\begin{corollary}[Detachment of slices and perturbed enclosures]
\label{cor:rpi:flow-detachment}
For any fixed $0\le s<t$, the flow frontiers $\Sigma_s$ and $\Sigma_t$ are
disjoint as closed perimeter supports. In either setting of this subsection, for
a $u$ satisfying the stated harmonic, boundary, strictness, and end hypotheses,
and for the forcing and sufficiently small fixed $\delta>0$ above, the
perimeter-plus-volume minimizer is disjoint from the original frontier, including
its singular set. Each fixed pair of these compact supports has positive
distance.
\end{corollary}
\begin{proof}
The flow assertion is the regular-contact contradiction plus
Proposition~\ref{prop:rpi:singular-separation}. For the perturbed enclosure, the
regular comparison above excludes regular contact, and the power
almost-minimizing estimate, the regular-patch bounds, and the absorbed graph
equation verify the hypotheses of
Proposition~\ref{prop:rpi:singular-separation}, which excludes singular contact.
Compactness of each pair of supports then gives positive distance. The constants
and the distance may depend on the fixed time pair or on the perturbation
parameter.
\end{proof}
\section{Capacity and one-sided smoothing}
\label{sec:rpi:orientation}
The mass decrease in conformal flow is controlled by the difference between ADM
mass and capacity. To see that the difference is nonnegative, we move a possibly
singular frontier toward the end and smooth it with a definite sign of mean
curvature. The capacity does not decrease under this operation, and then a smooth
mass--capacity inequality, proved from positive mass, applies.
Throughout this section $n\ge3$, $\omega=|S^{n-1}|$, and $\nu$ is the normal
from a compact removed region toward the asymptotic end. For a frontier $\Sigma$
we use $H_\nu=\divg_\Sigma\nu$. For a smooth compact conductor with exterior
$(E,h)$, normalize the capacity by
\begin{equation}\label{eq:rpi:capacity}
\mathfrak c(\Sigma,h)=\frac1{(n-2)\omega}\int_E|dw|_h^2\,dV_h,
\qquad \Delta_h w=0,\quad w|_\Sigma=1,\quad w\longrightarrow0 .
\end{equation}
Equivalently this is the normalized infimum of Dirichlet energy over smooth
compactly supported trial functions equal to one near the conductor; the same
variational definition applies to a nonsmooth conductor. If one conductor
contains another, its admissible class is smaller, so its capacity is no smaller.
This monotonicity is the whole reason the smoothing step helps.
\subsection{A harmonic perturbation and its detached enclosure}
Let $(E,g)$ be a flow-slice exterior or a smooth-ambient exterior as in
Theorem~\ref{thm:intro-main}, with nonempty frontier, capacitary potential $w$,
and $v=w-1$. The maximum principle gives $-1<v<0$ in the open exterior and
$v=0$ on the frontier. For $0<\varepsilon<1$ set
\[
u_\varepsilon=\frac{1+\varepsilon v}{1-\varepsilon},\qquad
\widetilde g_\varepsilon=u_\varepsilon^{4/(n-2)}g .
\]
Harmonicity gives $R_{\widetilde g_\varepsilon}=u_\varepsilon^{-4/(n-2)}R_g\ge0$,
and the factor tends to one at infinity. To fit the detachment corollary, whose
harmonic factor equals one on the boundary, write the same metric as
\[
g^{\rm base}_\varepsilon=(1-\varepsilon)^{-4/(n-2)}g,\qquad
\widehat u_\varepsilon=1+\varepsilon v,\qquad
\widetilde g_\varepsilon=\widehat u_\varepsilon^{4/(n-2)}g^{\rm base}_\varepsilon .
\]
Then $\widehat u_\varepsilon=1$ on the frontier and
$1-\varepsilon<\widehat u_\varepsilon<1$ in the exterior. Constant scaling
preserves harmonicity and local almost-minimality with changed constants. The
Hopf derivative of $\widehat u_\varepsilon$ in the direction $\nu$ is strictly
negative at every regular boundary point, so the conformal mean-curvature formula
makes the original frontier a strict inner barrier in $\widetilde g_\varepsilon$.
Now apply the prescribed-curvature construction of
Section~\ref{sec:rpi:perturbed-candidate} with a smooth forcing $h_0$ that is
zero on the original filled side and positive in its exterior. With a
sufficiently small fixed volume parameter, Corollary~\ref{cor:rpi:flow-detachment}
separates the candidate from the full original frontier. Denote its end exterior
by $E'$. A variation of its filled side by $\varphi\nu$ has first variation
\[
D\mathcal F[\varphi\nu]=\int_{\partial E'}(H_\nu+\delta h_0)\varphi\,dA,
\]
so its regular frontier satisfies $H_\nu=-\delta h_0\le-\gamma<0$ for some
$\gamma>0$, since the whole detached frontier lies in the original exterior where
$h_0>0$. Strictness matters: if $u\equiv1$, the forcing is zero and the original
exterior stays a minimizing competitor, so separation cannot follow from $u\le1$
alone.
\begin{remark}[Orientation]
On the Euclidean exterior $r\ge a$, our convention gives
\[
H_\nu=(n-1)/a>0,\qquad m_{\rm ADM}=0,\qquad
w=(a/r)^{n-2},\qquad\mathfrak c=a^{n-2}>0 .
\]
So the smooth mass--capacity inequality must be stated with the nonpositive sign
for this end-oriented normal. With the opposite (outward-exterior) normal the
sign flips, which is the convention some references use.
\end{remark}
\subsection{Smooth frontiers with the required sign}
\label{sec:rpi:smoothing}
The distance-offset construction of Bi--Zhu \cite{BiZhuSmoothing2026} supplies the geometric strategy for
the next proposition, but the conclusion we need is only conductor inclusion and
a mean-curvature bound. We prove exactly that, using Rifford's minimum-family
theorem locally and a final smoothing of transverse graphs. Boundary-area
convergence is not needed.
\begin{proposition}[One-sided smoothing in a shrinking collar]
\label{prop:rpi:one-sided-smoothing}
Let $\Omega$ be an open subset of a smooth Riemannian manifold with compact full
boundary $\Sigma$. Assume the metric is smooth on an ambient neighbourhood of
$\Sigma$. Let $\Lambda\subset\Sigma$ be the points touched by a geodesic ball
contained in $\Omega$. Suppose $\Lambda$ is a smooth embedded hypersurface,
locally bounds $\Omega$ on its accessible side, and has mean curvature at least
$\gamma>0$ for the normal pointing out of $\Omega$. Then for every
$\varepsilon>0$ there is an open set $\Omega_\varepsilon\subset\Omega$ with
smooth compact boundary such that
\[
\Omega\setminus\Omega_\varepsilon
\subset\{x\in\Omega:\dist(x,\Sigma)<\varepsilon\},\qquad
H_{\mathrm{out},\Omega_\varepsilon}\ge\gamma-\varepsilon .
\]
\end{proposition}
\begin{proof}
If $\Sigma$ is empty take $\Omega_\varepsilon=\Omega$. Otherwise choose a
regular distance level, take a short offset of its superlevel, and smooth the
resulting $C^{1,1}$ hypersurface; the offset controls curvature and the smoothing
loses an arbitrarily small amount. Write $\rho(x)=\dist(x,\Sigma)$ on $\Omega$
and fix $r_*>0$ far below the injectivity radii, so short paths stay in the
controlled region.
\emph{Regular distance levels.} If $0<\rho(x)<r_*$, every nearest point of
$\Sigma$ to $x$ belongs to $\Lambda$: a shortest segment from $x$ to the full
boundary stays in $\Omega$ and a terminal subsegment supplies an interior tangent
ball. Applying Rifford's theorem \cite{Rifford2004} on finitely many compact
smooth patches of $\Lambda$, the positive critical values of $\rho$ form a null
set; choose $s_i\downarrow0$ with $0\notin\partial_C\rho(x)$ whenever
$\rho(x)=s_i$, where $\partial_C$ is the Clarke differential.
\emph{Positive reach of the superlevels.} The distance function is locally
semiconcave away from $\Sigma$, with a uniform upper Hessian bound. At a regular
level let $\mathcal K=\partial_C\rho(x)$, a compact convex set not containing
zero. Separation of convex cones shows each unit proximal outward normal is
$\mathbf n=-p/|p|$ with $p\in\mathcal K$, and compactness gives $|p|\ge\eta>0$.
The semiconcavity inequality then yields
$\langle\mathbf n,\log_xy\rangle\le (C/\eta)\dist(x,y)^2$ for nearby $y$ of the
superlevel, which makes nearest-point projection unique for balls of radius
$a<\eta/(4C)$. Hence
$F_i=\{x\in\Omega:\rho(x)\ge s_i\}$ has positive reach; it is closed in the
ambient manifold and its boundary is compact.
\emph{Offsets and their curvature.} Fix a regular value $s=s_i$ and put $F=F_i$.
For $0<\tau<\operatorname{reach}(F)$ small compared with $s$, set
$O_\tau=\{x:\dist(x,F)<\tau\}$ and $\Gamma_\tau=\partial O_\tau$. The closure of
$O_\tau$ lies in $\Omega$ since $\rho\ge s-\tau>0$, and
$\Omega\setminus O_\tau$ lies in the $s$-collar of $\Sigma$. Each point of
$\Gamma_\tau$ is $\exp_q(\tau v)$ for a unique footpoint $q\in F$ and unit
outward normal $v$, and the same ray minimizes distance to $F$ for every
parameter below $\operatorname{reach}(F)$; the normal ray supplies tangent balls
on both sides, so $\Gamma_\tau$ is a compact $C^{1,1}$ hypersurface.
Transfer the positive curvature by comparison with smooth interior domains. Let a
smooth region $D\subset O_\tau$ touch $\Gamma_\tau$ from inside at
$p=\exp_q(\tau v)$. If the unit normal cone of $F$ at $q$ is a singleton, the
shortest segment from $q$ to $\Sigma$ has initial direction $v$ and endpoint
$z\in\Sigma$ on the accessible, smooth part. Enlarging $D$ by distance $s-\tau$
touches $\Sigma$ at $z$ on the $\Omega$ side, and Riccati comparison along the
touching geodesic gives $H(\text{enlarged }D\text{ at }z)\le H(D\text{ at }p)+
C_R(s-\tau)$, while first-contact comparison gives
$H(\text{enlarged }D\text{ at }z)\ge\gamma$. (If the enlargement is focal at $z$,
shrink $D$ by a small quadratic perturbation, apply Riccati, and let the
perturbation go to zero.) If the normal cone contains a direction not parallel
to $v$, convexity supplies a one-sided angular arc through $v$; exponentiating it
at radius $\tau$ and comparing second order gives a principal curvature at least
$1/(2\tau)$, while the exterior supporting ball bounds the others below, so
$H(D\text{ at }p)\ge 1/(2\tau)-(n-2)C_F$, which exceeds any fixed bound for
small $\tau$. Choose $s<\varepsilon$ with $C_Rs<\varepsilon/4$ and then $\tau$
small; the offset has outward mean curvature at least $\gamma-\varepsilon/4$ in
the barrier sense, hence its classical almost-everywhere mean curvature satisfies
the same bound.
\emph{Smooth graph approximation.} A compact cooriented embedded $C^{1,1}$
hypersurface whose almost-everywhere outward mean curvature is at least $c$ has
smooth embedded approximations converging in $C^1$ with mean curvature at least
$c-o(1)$. Choose a smooth ambient field $V$ uniformly transverse to the
hypersurface; the $C^1$ inverse function theorem and compactness give a fixed
flow tube in which each short flow line meets the hypersurface once. Projection
along these lines expresses the hypersurface as a flow graph of a $C^{1,1}$
function $u$ over a smooth nearby graph $\Gamma_0$. In local coordinates its
mean-curvature operator is $H(u)=A^{ab}(x,u,Du)\partial_{ab}u+B(x,u,Du)$ with
smooth coefficients. Convolution with normalized nonnegative kernels, weighted
by a partition of unity, converges to $u$ in $C^1$ with uniformly bounded second
derivatives, and the second derivatives become the partition-weighted averages of
$D^2u$ plus a uniformly vanishing term; since the coefficients are smooth on a
compact set of jets, the curvature sees the same average and
$H(u_\zeta)\ge c-o(1)$.
Apply this to $\Gamma_\tau$. Its positive distance from $\Sigma$ keeps a small
$C^1$ perturbation inside $\Omega$. Choose the region on the side containing the
deep part of $\Omega$; it differs from $O_\tau$ only in a tube around
$\Gamma_\tau$, of width smaller than both the distance to $\Sigma$ and
$\varepsilon-s$. So the removed collar lies in the $\varepsilon$-collar of
$\Sigma$, and choosing the smoothing small enough makes the curvature loss below
$\varepsilon/2$. The result is $\Omega_\varepsilon$.
\end{proof}
For the almost-minimizing frontiers used here, the accessible locus is exactly
the regular part: at a point touched by a ball in the exterior, a tangent cone
lies on one side of a plane, and small-excess regularity makes the point regular;
conversely a regular point has a small supporting ball on the chosen side.
Applying the proposition to a detached minimizing exterior $E'$ with
$H_\nu\le-\gamma$, its outward normal at the inner frontier is $-\nu$, so
$H_{\mathrm{out}}\ge\gamma$. With $\varepsilon=j^{-1}$ write
$\Omega_j=\Omega_{j^{-1}}$; since $\Omega_j\subset\Omega$, the new frontier lies
toward the end and satisfies $H_\nu(\partial\Omega_j)\le-\gamma+j^{-1}<0$. The new
conductor contains the old removed region, so by capacity monotonicity
$\mathfrak c(\partial\Omega_j,h)\ge\mathfrak c(\Sigma,h)$. Retaining only the
component containing the end preserves the sign and enlarges the conductor. This
smooth, complete exterior is what the mass--capacity lemma needs.
\section{Positive mass and the mass--capacity inequality}
\label{sec:rpi:smooth-repair}
The smoothing above reduced the singular-frontier argument to a smooth
mass--capacity inequality. We prove that inequality from a positive mass theorem,
independently of Theorem~\ref{thm:intro-main}. We use the smooth positive mass
result of Brendle--Wang \cite{BrendleWang2026}: a complete smooth
nonnegatively--scalar-curved manifold with a strict positive-scalar perturbation
and an all-order end expansion has nonnegative mass. The dimension-descent proof
of that theorem needs a covering estimate for weighted perimeter minimizers,
which Appendix~\ref{sec:rpi:weighted-minimizers} supplies. With that input
accounted for, the remaining steps are smooth: pass from strict to nonnegative
scalar curvature on harmonic ends, then double and repair the seam.
\subsection{Nonnegative mass on harmonic ends}
\begin{lemma}\label{lem:rpi:harmonic-pmt}
A smooth complete boundaryless manifold of dimension $n\ge8$ with finitely many
ends, all harmonically flat, and nonnegative scalar curvature has nonnegative
ADM mass at each end.
\end{lemma}
\begin{proof}
Work on the connected component containing the chosen end. Choose a positive
smooth function $\rho$, equal to $r^{-n-\beta}$ on each distant end with
$0<\beta<1$. The global Sobolev inequality on a manifold with finitely many
Euclidean ends, followed by exhaustion, gives the positive solution of
$-\Delta v=\rho$, $v\to0$ on every end. On each harmonically flat end $Uv$
satisfies a Euclidean Poisson equation with right side $-U^{(n+2)/(n-2)}\rho$, so
the Newton potential, split into inner, comparable, and outer annuli, gives
\[
v=Ar^{2-n}+O_j(r^{2-n-\beta'}),\qquad 0<\beta'<\beta .
\]
Thus $(1+tv)^{4/(n-2)}g$ is complete and has strictly positive scalar curvature
for every $t>0$, since
\[
-\frac{4(n-1)}{n-2}\Delta(1+tv)+R_g(1+tv)>0 .
\]
It satisfies the Brendle--Wang end expansion, so its ADM mass is nonnegative by
\cite{BrendleWang2026}. The mass changes by $2tA$ at the chosen end; letting
$t\downarrow0$ proves the claim.
\end{proof}
\subsection{Doubling and conformal repair}
The doubling turns the capacity of a smooth boundary into a mass decrease on a
complete manifold. It must keep the second end complete while positive mass is
applied; a positive parameter does this and can be sent to zero at the end.
\begin{lemma}[Mass--capacity inequality]
\label{lem:rpi:mass-capacity}
Let $(N,h)$, of dimension $n\ge8$, be a smooth connected exterior, complete
including its compact smooth boundary $\Gamma$, with exactly one end and compact
end complement. Suppose the end is harmonically flat and the scalar curvature is
nonnegative. Write $w$ for the capacitary potential, $c=\mathfrak c(\Gamma,h)$.
If $H_\nu\le0$ for the normal toward the end, then
$m_{\rm ADM}(h)\ge c$.
\end{lemma}
\begin{proof}
If $\Gamma$ is empty, $c=0$ and the claim is Lemma~\ref{lem:rpi:harmonic-pmt}.
Assume it is nonempty. Let $\phi=1-w$, so $\phi=0$ on $\Gamma$ and $\phi\to1$ at
the end. On the doubled manifold define $\psi=\phi$ on the first copy and
$\psi=-\phi$ on the second; across the seam $\psi$ and its first normal
derivative agree. For $0<a<1$ put
\[
V_a=\frac{1+a+(1-a)\psi}{2},\qquad h_a=V_a^{4/(n-2)}\bar h,
\]
where $\bar h$ is the doubled corner metric. Then $V_a\ge a$, tends to one at the
first end and to $a$ at the second, and is weakly harmonic, so $h_a$ is complete
with two harmonically flat ends and nonnegative scalar curvature away from the
seam. Taking the common transverse normal from the second copy to the first, the
two corner mean curvatures are $-H_\nu$ and $H_\nu$, so their jump is
$-2H_\nu\ge0$; the normal derivative of $V_a$ is continuous, so the conformal jump
has the same sign.
Smooth the seam locally by the Gaussian-collar construction of \cite{Miao2002},
which gives uniformly comparable $C^2$ metrics equal to $h_a$ off a shrinking
collar with a uniform negative-scalar bound, then approximate in $C^2$ by smooth
metrics $h_{a,\delta}$ that keep the bound. Choose a smooth $q_\delta\ge
(R_{h_{a,\delta}})_-$ supported in a slightly larger shrinking collar and solve
\[
\Delta_{h_{a,\delta}}z_\delta+\frac{n-2}{4(n-1)}q_\delta z_\delta=0,
\qquad z_\delta\longrightarrow1\ \text{at both ends}.
\]
Since $\|q_\delta\|_{n/2}\to0$, the associated bilinear form is coercive on the
homogeneous energy completion (Euclidean end inequalities plus a compact-core
Poincar\'e estimate with trace on one fixed end annulus), giving a smooth solution
$z_\delta\ge1$. The corrected metric $z_\delta^{4/(n-2)}h_{a,\delta}$ has
nonnegative scalar curvature and is harmonically flat at both ends. In separately
normalized end coordinates $u_\delta=z_\delta-1=A_{i,\delta}r_i^{2-n}+O(r_i^{1-n})$
with $A_{i,\delta}\ge0$, and flux integration over both ends gives
\[
(n-2)\omega\sum_{i=1}^2A_{i,\delta}\cdot\frac{4(n-1)}{n-2}
=\int q_\delta z_\delta\,dV\le\|q_\delta\|_1+C\|q_\delta\|_p^2=o(1),
\]
so the end masses converge to those of $h_a$. Lemma~\ref{lem:rpi:harmonic-pmt}
gives nonnegative limiting mass. On the retained end
\[
\phi=1-cr^{2-n}+O_j(r^{1-n}),\qquad
V_a=1-\tfrac12(1-a)cr^{2-n}+O_j(r^{1-n}),
\]
so the conformal ADM formula gives $0\le m_{\rm ADM}(h_a)=m_{\rm ADM}(h)-(1-a)c$.
Letting $a\downarrow0$ proves $m_{\rm ADM}(h)\ge c$.
\end{proof}
The signed variational construction, one-sided smoothing, and
Lemma~\ref{lem:rpi:mass-capacity} now give the smooth comparison needed for a
singular time slice: an enclosing smooth conductor has at least the original
capacity, while moving the frontier leaves the end mass unchanged.
Section~\ref{sec:rpi:flow-mass-capacity} applies this to a small harmonic
perturbation and removes it, and the same positive two-ended construction supplies
the complete inputs used in Section~\ref{sec:rpi:lee-completion}.
\section{Mass and capacity along the flow}
\label{sec:rpi:flow-mass-capacity}
The smooth comparison now applies to every flow slice after a small harmonic
perturbation. Together with the end expansion it gives the mass decrease that
drives the whole argument.
\begin{proposition}[Mass and capacity of the conformal flow]
\label{prop:rpi:mass-evolution}
For the conformal flow of Lemma~\ref{lem:rpi:flow-construction} in dimension
$n\ge8$, with nonempty initial frontier and harmonically flat initial end, let
$m(t)$ be the ADM mass in normalized coordinates and $c_t=\mathfrak c(\Sigma_t,g_t)$.
Then $m(t)\ge c_t\ge0$; with $\mu(t)=m(t)-c_t\ge0$, the mass is locally absolutely
continuous and
\begin{equation}\label{eq:rpi:mass-evolution}
m'(t)=-2\mu(t)\qquad\text{almost everywhere.}
\end{equation}
\end{proposition}
\begin{proof}
\emph{Nonnegativity of the deficit.} Fix a slice with normalized metric $h$ and
write $m,c,w$ for its mass, capacity, and capacitary potential in normalized end
coordinates. For small $\lambda>0$ set $p=1+\lambda w$ and
$h_\lambda=p^{4/d}h$. Conformal harmonicity and the end expansions give
\[
w_{\Sigma,\lambda}=\frac{(1+\lambda)w}{p},\qquad
c_{\Sigma,\lambda}=(1+\lambda)c,\qquad
m_\lambda=m+2\lambda c .
\]
This is the separating perturbation of
Section~\ref{sec:rpi:orientation} with $\lambda=\varepsilon/(1-\varepsilon)$. Its
detached smooth enclosure $\Gamma$ has $H_\nu<0$, so capacity monotonicity and
Lemma~\ref{lem:rpi:mass-capacity} applied to its smooth exterior give
\[
m+2\lambda c=m_\lambda\ge c_\Gamma\ge(1+\lambda)c .
\]
Letting $\lambda\downarrow0$ proves $m\ge c$.
\emph{The evolution formula.} Write the initial end as $g_0=H^{4/d}\delta$ with
$H>0$, $\Delta_\delta H=0$, $H\to1$. The products $Hu_t$ and $Hv_t$ are Euclidean
harmonic on the end, so expanding and dividing by $H$ gives, for $0<\beta<1$,
\[
u_t=e^{-t}+B(t)r^{-d}+O_1(r^{-d-\beta}),\quad
v_t=-e^{-t}+b(t)r^{-d}+O_1(r^{-d-\beta}),\quad B'=b,\ B(0)=0 .
\]
Flux identification and normalization of the end coordinates give
\[
c_t=e^{-t}(B+b),\qquad m(t)=e^{-2t}m(0)+2e^{-t}B(t),\qquad
\mu(t)=m(t)-c_t .
\]
Since $B$ is locally absolutely continuous, so is $m$, and differentiating gives
\eqref{eq:rpi:mass-evolution}.
\end{proof}
\subsection{The normalized end factors}
Let $\Phi_t(x)=e^{2t/d}x$ map the normalized coordinate $x$ to the original one.
The normalized flow metric is $G_t=\Phi_t^*g_t$ on this end. Transport the flow
and velocity factors by
\[
U_t=e^tu_t\circ\Phi_t,\qquad V_t=e^tv_t\circ\Phi_t,\qquad
h_t=H\circ\Phi_t,\qquad \bar G_t=e^{-4t/d}\Phi_t^*g_0=h_t^{4/d}\delta .
\]
The velocity $V_t$ is $\bar G_t$-harmonic on the part of the normalized exterior
in this chart. Define the total factors and the auxiliary relative factor:
\[
\begin{array}{c|c|c}
\text{role}&\text{relative factor}&\text{Euclidean-harmonic factor}\\ \hline
\text{flow metric}&U_t&F_t=h_tU_t\\
\text{velocity}&V_t&Z_t=h_tV_t\\
\text{auxiliary metric}&W_t=(U_t-V_t)/2&J_t=h_tW_t
\end{array}
\]
Then $G_t=F_t^{4/d}\delta$ and
\begin{equation}\label{eq:rpi:auxiliary-factor}
J_t=\frac{F_t-Z_t}{2}=F_t\Bigl(1-\frac{w_t\circ\Phi_t}{2}\Bigr),\qquad
\widetilde G_t=J_t^{4/d}\delta .
\end{equation}
The total factors are harmonic outside a sphere enclosing the current frontier and
compact core. Their constant terms are $1,-1,1$, and the masses of the flow and
auxiliary metrics are $m(t)$ and $\mu(t)$.
Section~\ref{sec:rpi:lee-completion} constructs complete smooth metrics whose end
factors approximate $J_t$, which lets a quantitative positive-mass estimate
control it.
\section{Vanishing deficit and the enclosure radius}
\label{sec:rpi:flow-estimates}
Area preservation and Proposition~\ref{prop:rpi:mass-evolution} reduce the
argument to the decay of $\mu(t)=m(t)-c_t$ and the location of the frontier. The
next estimates provide a fixed harmonic tail in normalized coordinates on which
the quantitative positive-mass theorem will apply. Put $q=2(n-1)/d$ and retain
$\omega=|S^{n-1}|$.
\begin{proposition}[Vanishing deficit and bounded normalized radius]
\label{prop:rpi:flow-estimates}
For the flow in Proposition~\ref{prop:rpi:mass-evolution}, $\mu(t)\to0$ as
$t\to\infty$. There is a fixed $R_{\max}$ such that
\begin{equation}\label{eq:rpi:radius-bound}
\Sigma_t\subset B_{R_{\max}e^{2t/d}}\qquad(t\ge0),
\end{equation}
the radius measured in the original end coordinates with the compact core
included. Equivalently, every normalized frontier lies in $B_{R_{\max}}$.
\end{proposition}
\begin{proof}
\emph{The integrated mass identity.} Since $B'=b$ and $B(0)=0$,
$\frac{d}{dt}(e^{-t}B(t))=e^{-t}(b(t)-B(t))$, so
\[
\int_0^t\mu(s)\,ds
=\frac12(1-e^{-2t})m(0)-e^{-t}B(t)=\frac{m(0)-m(t)}2 .
\]
This supplies integrability of the nonnegative $\mu$. Set $a=m-2\mu=2c_t-m$.
Nesting and the maximum principle make $e^tv_t$ nondecreasing, and its end
coefficient $e^tb(t)$ is therefore nondecreasing; since
$e^{2t}a(t)=2e^tb(t)-m(0)$, the same holds for $e^{2t}a(t)$. This implies
$Da+2a\,dt\ge0$ distributionally, and substitution of the integral identity makes
$3m-2\mu+2\int_0^tm$ monotone. Hence for $h>0$,
\[
\mu(t+h)-\mu(t)\le\frac32\bigl(m(t+h)-m(t)\bigr)+\int_t^{t+h}m(s)\,ds\le h\,m(0).
\]
This one-sided slope bound together with integrability forces $\mu(t)\to0$: a
value at least $\varepsilon>0$ would force an interval immediately before it on
which the integral is bounded below by a positive constant depending only on
$\varepsilon,m(0)$. If $m(0)=0$, nonnegativity and mass monotonicity give $m=0$,
and the slope bound excludes positive point values of $\mu$.
\emph{The escaping-point ball.} Put $\lambda=e^{2/d}$, $R(s)=R_{\max}\lambda^s$.
Suppose containment holds for $0\le s\le T$ but there is $p\in\Sigma_{T+1}$ with
$|p|\ge\lambda R(T)$. Use the ball $\rho=\frac{\lambda-1}{2}R(T)$ with
$\overline{B_\rho(p)}$ in $\{|y|>R(T)\}$ in the original end coordinates. Choose
$R_{\max}$ so that exterior avoids the initial obstacle and
$\tfrac12\delta\le g_0\le2\delta$ there.
Write $g_0=H^{4/d}\delta$ on the initial end with $H$ Euclidean harmonic,
$H\to1$. Then $Hv_s$ is Euclidean harmonic outside $\Sigma_s$; if
$\Sigma_s\subset B_{R(s)}$ its boundary value on $S_{R(s)}$ is nonpositive and its
limit at infinity is $-e^{-s}$, so
\[
Hv_s(y)\le e^{-s}\Bigl[\Bigl(\frac{R(s)}{|y|}\Bigr)^d-1\Bigr]\qquad(|y|\ge R(s)).
\]
Using $v_s\le0$ on the remaining time interval gives, for $r=|y|\ge R(T)$,
\[
Hu_{T+1}\le H-1+e^{-T}+(e^T-1)R_{\max}^d r^{-d}.
\]
Choosing $H-1\le C_0r^{-d}$ and $R_{\max}^d\ge C_0$, the right side is at most
$2e^{-T}$; with $H\ge\tfrac12$ this gives the concrete bounds
\[
e^{-T-1}\le u_{T+1}\le Ke^{-T},\quad K=4,\qquad r\ge R(T).
\]
Every finite-perimeter variation supported in $B_\rho(p)$ remains an admissible
enclosure, so comparison of area elements makes $\Sigma_{T+1}$ Euclidean
perimeter-quasiminimizing there with ratio $\gamma=2^{n-1}(eK)^q$. The
isoperimetric density argument then gives
$\mathcal H_\delta^{n-1}(\Sigma_{T+1}\cap B_\rho(p))\ge c(n,\gamma)\rho^{n-1}$.
So the fixed area satisfies
\[
A\ge2^{-(n-1)/2}e^{-q(T+1)}c(n,\gamma)\rho^{n-1}
=2^{-(n-1)/2}c(n,\gamma)e^{-q}
\Bigl(\frac{\lambda-1}{2}\Bigr)^{n-1}R_{\max}^{n-1}.
\]
All time factors cancel (since $q\,d/2=n-1$). This is a fixed lower bound growing
like $R_{\max}^{n-1}$ with a coefficient independent of $T$ and of further
enlargement of $R_{\max}$, contradicting the fixed value of $A$ once $R_{\max}$ is
large. Enlarge $R_{\max}$ to get containment on $[0,1]$ from the finite-time
support bound; then apply the argument at each $t\in[N,N+1]$ with $T=t-1$ to get
\eqref{eq:rpi:radius-bound} at every real time.
\end{proof}
\section{Complete metrics and the auxiliary end factor}
\label{sec:rpi:lee-completion}
Lee's quantitative positive-mass estimate turns small mass into control of a
harmonic end factor. To apply it to the current slice we construct complete
smooth metrics whose masses tend to zero and whose end factors approach $J_t$.
The construction keeps a fixed coordinate tail while allowing the compact
interior to vary. Throughout this section $n\ge8$ and $d=n-2$.
\begin{proposition}[Complete approximants and the auxiliary factor]
\label{prop:rpi:complete-end-factors}
For the normalized flow of Proposition~\ref{prop:rpi:mass-evolution} there are
fixed radii $R_c<R_1$ and complete smooth connected boundaryless one-ended
metrics with nonnegative scalar curvature, harmonic end factors $\mathcal Z_t$ on
$\{r>R_1\}$, and masses $\mathcal M_t$, such that
\[
0\le\mathcal M_t\longrightarrow0,\qquad
\sup_{r\ge2R_1}r^d|\mathcal Z_t-J_t|\longrightarrow0 .
\]
The radius $R_c$ encloses every normalized frontier and compact core. For every
fixed $s>R_c$,
\begin{equation}\label{eq:rpi:auxiliary-weighted-limit}
\sup_{r\ge s}r^d|J_t-1|\longrightarrow0 .
\end{equation}
\end{proposition}
The quantitative input is isolated first, so the construction can be read
separately from its positive-mass hypotheses.
\begin{lemma}[Quantitative positive mass on one harmonic end]
\label{lem:rpi:quantitative-one-end}
For each $n\ge8$ there is a fixed constant $\ell_n>1$ such that, for every
$\eta>0$, there is $\delta(n,\eta)>0$ with the following property. Let $(N,h)$ be
a smooth complete connected boundaryless $n$-manifold with $R_h\ge0$. Suppose
that outside a compact set it is a single coordinate end $\{r>R\}$ on which
$h=Z^{4/(n-2)}\delta$, $Z>0$, $\Delta_\delta Z=0$, $Z\to1$. Writing
$m=m_{\rm ADM}(h)$, if $m<\delta(n,\eta)R^d$ then
$\sup_{r>\ell_n R}|Z-1|<\eta$.
\end{lemma}
\begin{proof}
Rescale by $R^{-2}$, so the harmonic coordinate radius becomes one and the mass
becomes $m/R^d$. Apply the argument of \cite[Theorem~1.3]{Lee2007NearEquality} at
this scale. Its positive-mass hypotheses are supplied here by
Lemma~\ref{lem:rpi:harmonic-pmt}, and the one-end assumption identifies the mass
flux in Lee's Lemma~2.1 with the flux over the whole manifold. The scalar-flat
reduction of \cite[Lemma~2.5]{Lee2007NearEquality} gives $k=f^{4/d}h$, $f>0$,
$f\to1$, bounded above and below, so $k$ stays smooth and complete, and its end
factor $Zf$ is Euclidean harmonic. The variation and conformal correction
(Lee's Lemma~2.4 with the correct fixed dimensional coefficient) gives
$u_s-1=O(|s|)$ uniformly, and $1/2\le u_s\le3/2$ for small $|s|$, so the
varied metrics stay smooth and complete. Nonnegativity of the varied mass, again
from Lemma~\ref{lem:rpi:harmonic-pmt}, gives Lee's Lemma~2.2 input and
$m_{\rm ADM}(k)\le m_{\rm ADM}(h)$. The constants depend only on dimension and
the target error at the normalized radius, not on the compact interior. Restoring
the scale $R$ gives the stated implication.
\end{proof}
\subsection{From two ends to one}
We use an elementary harmonic estimate. If $q\ge0$ is Euclidean harmonic on
$\{r>R\}$, tends to zero, and has expansion $q=br^{-d}+O(r^{-d-1})$, then
\begin{equation}\label{eq:rpi:positive-harmonic-monopole}
0\le q(x)\le C(n)b\,r^{-d}\qquad(r\ge2R):
\end{equation}
the spherical average is exactly $br^{-d}$, and the Harnack inequality on scaled
annuli bounds the maximum on a sphere by a dimensional constant times this
average, independently of the geometry inside $B_R$.
\emph{Discarding a second end.} Let $(N,k)$ be smooth complete connected
boundaryless with exactly two harmonically flat ends and compact end complement,
$R_k\ge0$. Designate one end for retention. There is a harmonic $\chi$ tending to
one on that end and zero on the other with $0<\chi<1$: take a smooth function
with these exterior values and minimize Dirichlet energy after adding the
homogeneous energy space. Write $k=Z_i^{4/d}\delta$ on either end; then $Z_i\chi$
is Euclidean harmonic, and on the end to be filled
$Z_2\chi=b_2r^{-d}+O_j(r^{-d-1})$ with $b_2>0$. Under inversion
$x=y/|y|^2$, the factor $|y|^{-d}(Z_2\chi)(y/|y|^2)$ extends harmonically across
$y=0$ with value $b_2$, so $\chi^{4/d}k$ extends to a smooth metric after adding
one point. The filled manifold is complete, has one end, and has nonnegative
scalar curvature. At the retained end write $\chi=1-Ar^{-d}+O_j(r^{-d-1})$; if
$M=m_{\rm ADM}(k)$, the filled metric has mass $M-2A$. The function
$Z_1(1-\chi)$ is nonnegative Euclidean harmonic with monopole $A$, so
\begin{equation}\label{eq:rpi:one-end-reduction}
0\le M-2A\le M,\qquad |Z_1-Z_1\chi|\le C(n)M r^{-d}\quad(r\ge2R),
\end{equation}
the first inequality by Lemma~\ref{lem:rpi:harmonic-pmt} on the filled manifold.
The estimate depends only on the retained-end radius and mass; the second end may
vary with the compact interior.
\subsection{Completion, capacity, and the auxiliary factor}
\emph{A common coordinate tail.} Use the normalized metric $G_t$ and total
harmonic factor $F_t$ of Section~\ref{sec:rpi:flow-mass-capacity}; in this chart
$\Sigma_t,w_t$ denote the frontier and its capacitary potential (the pullback of
the potential previously written $w_t\circ\Phi_t$). A coordinate ball includes the
compact core when the whole exterior is under discussion. The radius bound of
Proposition~\ref{prop:rpi:flow-estimates} and the harmonic estimates give
$\Sigma_t\subset B_{R_c}$, the factors $F_t$ uniformly bounded on a fixed sphere
outside $B_{R_c}$, and
\[
0\le c_t\le m(t)\le m(0),\qquad \mu(t)\longrightarrow0 .
\]
Indeed, taking $T=t-1$ in the radius estimate gives $U_t\le4e$ on the normalized
tail $r\ge R_{\max}$ for $t\ge1$; the initial bounded time interval is covered by
finite-time flow bounds; multiplication by the uniformly bounded initial harmonic
factor gives the bound for $F_t$. Let $w_t$ be the capacitary potential equal to
one at $\Sigma_t$ and zero at infinity.
For $0<\lambda<1$ put $p=1+\lambda w_t$ and $h_{t,\lambda}=p^{4/d}G_t$. The
detachment and smoothing construction above provides a smooth enclosing boundary
$\Gamma$ with $H_\nu^\Gamma<0$, chosen in a ball $B_{R_1}$ whose radius is
independent of $t$ and $\lambda$. The confinement is as follows. On the common
tail $F_tp$ is positive and harmonic, uniformly bounded on $S_{R_c}$, and tends to
one; the maximum principle and scaled gradient estimates give
\[
|F_tp-1|+r|D(F_tp)|\le C(R_c/r)^d\qquad(r\ge2R_c),
\]
so all coordinate spheres beyond a fixed $R_*$ have positive outward mean
curvature in $h_{t,\lambda}$. Clipping a minimizer $L$ of the perimeter-plus-volume
functional by such a sphere decreases both perimeter and the nonnegative volume
penalty, so $L\subset B_{R_*}$; the subsequent smooth boundaries lie in shrinking
collars, hence in $B_{R_1}$. Thus the coordinate tail is fixed even as the
separation and smoothing parameters vary.
\emph{Capacity comparison.} Suppress $t$, putting $h=G_t$, $F=F_t$, $w=w_t$,
$c=c_t$, $m=m(t)$, $\mu=\mu(t)$. The potential and capacity of the original
conductor for $h_\lambda=p^{4/d}h$ are exactly
\[
w_{\Sigma,\lambda}=\frac{(1+\lambda)w}{p},\qquad
c_{\Sigma,\lambda}=(1+\lambda)c,\qquad m_\lambda=m+2\lambda c .
\]
If $w_\Gamma$ is the potential of $\Gamma$ for $h_\lambda$ and $c_\Gamma$ its
capacity, enclosure monotonicity and Lemma~\ref{lem:rpi:mass-capacity} give
\[
0\le D_\Gamma:=c_\Gamma-c_{\Sigma,\lambda}\le m_\lambda-c_{\Sigma,\lambda}
=\mu+\lambda c .
\]
On the common tail $Fp(w_\Gamma-w_{\Sigma,\lambda})$ is nonnegative Euclidean
harmonic with monopole $D_\Gamma$.
\emph{The doubling end factor.} For $0<a<1$ the positive two-ended doubling
construction of Lemma~\ref{lem:rpi:mass-capacity} has retained end factor and
retained end mass
\[
B=Fp\Bigl(1-\frac{1-a}{2}w_\Gamma\Bigr),\qquad
m_B=m_\lambda-(1-a)c_\Gamma\le\mu+\lambda c+a m_\lambda .
\]
The Miao smoothing and its conformal correction produce smooth complete two-ended
metrics with $R\ge0$ and retained factor $Bz_\delta$ on $\{r>R_1\}$; here
$z_\delta\ge1$ and $e_\delta:=B(z_\delta-1)\ge0$ is Euclidean harmonic with
monopole $A_\delta\to0$, and the retained masses are $M_\delta=m_B+2A_\delta\ge0$.
The auxiliary Euclidean end factor is $J=F(1-w/2)$. Direct subtraction gives
\[
B-J=\frac{\lambda}{2}Fw-\frac12Fp(w_\Gamma-w_{\Sigma,\lambda})
+\frac a2Fp\,w_\Gamma,
\]
with each harmonic function nonnegative before its displayed sign and monopoles
$c$, $D_\Gamma$, $c_\Gamma$. Equation~\eqref{eq:rpi:positive-harmonic-monopole}
therefore yields
\begin{equation}\label{eq:rpi:completed-factor-comparison}
|Bz_\delta-J|\le C(n)\bigl(\mu+\lambda c+a m_\lambda+A_\delta\bigr)r^{-d},
\qquad r\ge2R_1 .
\end{equation}
The capacity difference controls the end-factor error independently of the
distance between $\Gamma$ and $\Sigma_t$.
For each $t$ choose $\lambda_t=a_t=(t+2)^{-1}$, then the detachment and smoothing
parameters, and finally $\delta_t$ with $A_{\delta_t}\le(t+2)^{-1}$. The preceding
bounds give $0\le M_{\delta_t}\to0$ and
$\sup_{r\ge2R_1}r^d|B_tz_{\delta_t}-J_t|\to0$. Apply the one-end reduction
\eqref{eq:rpi:one-end-reduction} to each completed metric: the double is connected
and has exactly two ends, so it yields smooth connected one-ended metrics with
masses $\mathcal M_t$ and harmonic end factors $\mathcal Z_t$, on the same
$\{r>R_1\}$, with $\mathcal M_t\to0$ and
$\sup r^d|\mathcal Z_t-J_t|\to0$. Lemma~\ref{lem:rpi:quantitative-one-end},
followed by this end-factor comparison, gives
\[
J_t\longrightarrow1\quad\text{uniformly on }\{r>R_2\}
\]
for any fixed $R_2>\max(2,\ell_n)R_1$.
\emph{The inner tail.} The initial harmonic factor in normalized coordinates is
$h_t=H\circ\Phi_t$, so $\bar G_t=h_t^{4/d}\delta$, $F_t=h_tU_t$, $J_t=h_tW_t$ with
$W_t=\frac12(U_t-V_t)$. The initial end expansion gives
$h_t-1=O_j(e^{-2t}r^{-d})$ on any fixed exterior tail, so the preceding conclusion
gives $W_t\to1$ there. To extend to each fixed radius $s>R_c$: the functions
$J_t$ are positive Euclidean harmonic on $\{r>R_c\}$ with spherical averages
$1+\frac12\mu(t)r^{-d}$. Bounded masses give uniform Harnack and elliptic bounds
on each compact subannulus, and every subsequential harmonic limit equals one on
the distant tail, hence everywhere on $\{r>R_c\}$ by unique continuation. Thus
$J_t\to1$ uniformly on $S_s$, and the maximum principle applied to $J_t-1$
outside $S_s$ yields
\[
\sup_{r\ge s}r^d|J_t-1|\le s^d\sup_{S_s}|J_t-1|\longrightarrow0 .
\]
The same holds for $W_t$ since $h_t\to1$ with that decay. This proves
\eqref{eq:rpi:auxiliary-weighted-limit}.
\section{The end limit and the numerical inequality}
\label{sec:rpi:numerical-limit}
The vanishing deficit determines a limiting radial potential. Its zero boundary
value identifies the largest limiting frontier radius, and comparison with
enclosing spheres gives the sharp full-area bound.
\begin{proposition}[The inequality for a harmonically flat end]
\label{prop:rpi:harmonic-flat-inequality}
Let $(E,h)$ satisfy the hypotheses of Theorem~\ref{thm:intro-main}, with $n\ge8$,
nonempty frontier of full area $A$, and a harmonically flat end. Then
\[
m_{\rm ADM}(h)\ge\frac12\Bigl(\frac A\omega\Bigr)^{d/(n-1)} .
\]
\end{proposition}
\begin{proof}
The conformal flow preserves $A$ and has decreasing mass $m(t)\to M\ge0$.
Proposition~\ref{prop:rpi:complete-end-factors} controls its auxiliary factor on a
common tail. Put $q=2(n-1)/d$.
\emph{Characteristic equation.} The flow metric has factor $F_t$ and the
auxiliary metric factor $J_t$, with monopole coefficients $m(t)/2$ and $\mu(t)/2$.
The background factor $h_t=H\circ\Phi_t$ contributes the initial mass: the $r^{-d}$
coefficient of $U_t$ is $e^{-t}B(t)$, while that of $F_t$ is
$m(t)/2=e^{-t}B(t)+e^{-2t}m(0)/2$. Set
\[
f_t=F_t-1-\frac{m(t)}2r^{-d},\qquad
j_t=J_t-1-\frac{\mu(t)}2r^{-d}.
\]
The evolution and mass identities give
$\partial_tf_t=2(f_t-j_t)+\frac2d r\partial_rf_t$, so for
$y_s=e^{2(t-s)/d}x$ the chain rule gives
\begin{equation}\label{eq:rpi:lim-characteristic}
\frac{d}{ds}\bigl(e^{-2s}f_s(y_s)\bigr)=-2e^{-2s}j_s(y_s).
\end{equation}
\emph{A quantitative bound for $j_t$.} Equation
\eqref{eq:rpi:auxiliary-weighted-limit} gives
$\sup_{r\ge s_0}r^d|J_t-1|\to0$ for every fixed $s_0>R_c$. Choose
$s_0=2\max\{R_c,R_{\max},1\}$. The Kelvin transform
\[
K_t(y)=|y|^{-d}\bigl(J_t(y/|y|^2)-1\bigr)
\]
extends harmonically across zero, has value $K_t(0)=\mu(t)/2$, and is uniformly
small on $|y|\le s_0^{-1}$. The interior gradient estimate on the half ball gives
$|j_t(x)|\le C\varepsilon r^{-d-1}$ for all large $t$ and $r\ge2s_0$. Hence for
$0<\beta\le1$, $s\ge t_0$, $r\ge2s_0$,
$|j_s(x)|\le C\varepsilon r^{-d-\beta}$ and $|f_{t_0}(x)|\le C_0r^{-d-\beta}$.
Integrating \eqref{eq:rpi:lim-characteristic} gives
\[
|f_t(x)|\le\Bigl[C_0e^{-2\beta(t-t_0)/d}
+\frac{dC}{\beta}\varepsilon\bigl(1-e^{-2\beta(t-t_0)/d}\bigr)\Bigr]r^{-d-\beta}.
\]
The first term tends to zero with $t$ fixed $t_0$; taking $t\to\infty$ and then
$\varepsilon\downarrow0$ proves the exterior convergence of the total factors:
\[
F_t\longrightarrow1+\frac M2r^{-d},\qquad
Z_t=F_t-2J_t\longrightarrow-1+\frac M2r^{-d}.
\]
Since $h_t\to1$ on any fixed exterior region, $V_t=Z_t/h_t$ has the latter limit.
\emph{Boundary control on fixed annuli.} Choose any sequence $t_i\to\infty$. On
the end chart use $x=e^{-2t_i/d}y$; let $E_i,\Sigma_i$ be the rescaled exterior
and frontier, and let $\mathcal C_i$ be the compact set formed by adjoining to the
rescaled boundary portion the closed ball of radius $R_{\rm chart}e^{-2t_i/d}$
representing the collapsing core. The radius bound puts $\mathcal C_i$ in a fixed
ball. On any fixed compact annulus away from the origin, the rescaled upper and
lower conformal bounds give uniform comparison of $G_{t_i}$ with $\delta$: with
$L\ge\max\{1,\frac d2\log(R_{\max}/a_0)\}$ and $S=t_i-L\ge0$, if $|x|\ge a_0$ then
$|y|\ge R(S)$, and monotonicity of $u_t$ gives
\[
e^{-t_i}\le u_{t_i}(y)\le u_{S+1}(y)\le4e^{-S},\qquad 1\le U_{t_i}(x)\le4e^L .
\]
The rescaled initial obstacle is eventually disjoint from the annulus, so the
enclosure property permits both filling and deleting local balls, giving a uniform
Euclidean quasiminimality ratio and both-phase density
$|B_s(p_i)\cap E_i|\ge\eta s^n$, $|B_s(p_i)\setminus E_i|\ge\eta s^n$ for
$p_i\in\Sigma_i$, $0<s<\rho_0$. The backgrounds $\bar G_{t_i}$ converge smoothly
to $\delta$ on the annulus, so the operators for $V_i$ are uniformly elliptic.
Construct the background equilibrium potential and write $\mathcal P_i$ for its
pullback; then $q_i=1-\mathcal P_i$ has a $W^{1,2}_{\rm loc}$ zero extension, is a
weak subsolution, and satisfies $0\le q_i\le1$. The weak Harnack inequality gives
$\sup_{B_s(p_i)}q_i\le(1-\theta)\sup_{B_{4s}(p_i)}q_i$, and iteration yields
\[
|q_i(x)|\le C\Bigl(\frac{|x-p_i|}{\rho_0}\Bigr)^\kappa
\qquad(|x-p_i|<\rho_0),\quad\kappa>0 .
\]
The flow potential satisfies $q_i=-V_i$ on the exterior portion, so this controls
$|V_i|$ uniformly, including at singular frontier points.
\emph{The largest limiting radius.} Take a Hausdorff limit $\mathcal C$ of
$\mathcal C_i$, and let $O$ be the unbounded component of $\R^n\setminus\mathcal C$.
Every compact subset of $O$ eventually lies in the actual exterior. The uniform
annular bounds and interior elliptic estimates give a subsequence of $F_{t_i},V_i$
converging on compact subsets of $O$, and the far-exterior limits identify both
functions throughout $O$ by unique continuation:
\[
F_\infty(x)=1+\frac M2|x|^{-d},\qquad V_\infty(x)=-1+\frac M2|x|^{-d},
\]
with $M=\lim m(t)$. The origin lies in $\mathcal C$, hence outside $O$. The
boundary H\"older estimate gives $V_\infty=0$ at every point of
$\partial O\setminus\{0\}$. Set $R_*=\max\{|x|:x\in\mathcal C\}$. If $R_*>0$,
choose $p\in\mathcal C$ with $|p|=R_*$; points on the outward radial ray from $p$
lie in $O$, so the boundary estimate and the displayed potential give
\[
0=-1+\frac M{2R_*^d},\qquad R_*=r_\infty:=(M/2)^{1/d}.
\]
In particular $M>0$ and $\mathcal C\subset\overline B_{r_\infty}$.
If $R_*=0$ then $\mathcal C=\{0\}$ and the same harmonic limit holds on every fixed
annulus. Its sign $V_\infty\le0$ forces $M=0$. For any fixed $R>0$ the normalized
boundaries are eventually in $B_R$, while the total factors converge to $1$ on
$S_R$ by harmonic compactness and unique continuation. The minimizing-enclosure
property would then give $A\le\omega R^{n-1}$ for every $R>0$, contradicting
$A>0$. Thus $R_*>0$ and $M>0$.
\emph{The area comparison.} For any $R>r_\infty$, outer containment and the
minimizing property give, after taking the subsequential limit,
\[
A\le\omega R^{n-1}\Bigl(1+\frac M{2R^d}\Bigr)^q .
\]
Letting $R\downarrow r_\infty$ yields
\[
A\le\omega(2M)^{(n-1)/d},\qquad
m(0)\ge M\ge\frac12\Bigl(\frac A\omega\Bigr)^{d/(n-1)} .
\]
\end{proof}
\section{Passage from general ends to harmonic-flat ends}
\label{sec:rpi:density-repair}
The harmonic-flat argument of the previous sections assumed an end of the form
$H^{4/d}\delta$. A general asymptotically flat end is only close to Euclidean to
some order. This last section removes the assumption. It first constructs a strict
smooth inner barrier so that the detached frontier sits in a smooth neighborhood,
then approximates the metric by harmonically flat metrics, controlling the mass and
every enclosing area, and finally takes a detached minimizing enclosure. The
approximation never needs a theorem that preserves a singular minimal frontier.
\begin{lemma}[Smooth exterior harmonic-flat approximation]
\label{lem:rpi:smooth-density}
Let $(N,h)$ be a smooth connected asymptotically flat exterior, complete including
its compact smooth boundary $\Gamma$ (possibly empty), with one coordinate end and
compact end complement. Suppose, with $d=n-2$,
\[
h-\delta=O_2(r^{-\tau}),\quad\tau>\tfrac d2,\qquad
R_h\ge0,\quad R_h\in L^1,\quad R_h=O(r^{-n-\beta}),\quad\beta>0,
\]
and the ADM mass exists. Then there are smooth metrics $\widehat h_R$, complete with
the same boundary, with nonnegative scalar curvature and a harmonically flat end,
such that
\[
(1-\epsilon_R)h\le\widehat h_R\le(1+\epsilon_R)h,\quad\epsilon_R\to0,\quad
\widehat h_R\to h\ \text{in }C^2\ \text{on every fixed compact set},\quad
m_{\rm ADM}(\widehat h_R)\to m_{\rm ADM}(h).
\]
If $H_\nu(\Gamma,h)<0$ for the normal toward the end, the same strict sign holds
for $\widehat h_R$ when $R$ is large.
\end{lemma}
\begin{proof}
Cut off the metric on a distant annulus and remove the resulting scalar-curvature
error by a Dirichlet conformal correction; the annular flux then recovers the
original ADM mass.
\emph{The cutoff and Dirichlet correction.} Choose a smooth radial cutoff
$\chi_R=\chi(r/R)$ with $\chi=1$ on $(-\infty,1]$, $0$ on $[2,\infty)$,
$0\le\chi\le1$. On the end put $h_R=\delta+\chi_R(h-\delta)$, and set $h_R=h$ on
the compact core; for large $R$ this is a smooth positive metric equal to $h$ on
$r\le R$ and Euclidean on $r\ge2R$. Define $Q_R=R_{h_R}-\chi_RR_h$, supported in
the annulus $\mathcal A_R=\{R<r<2R\}$. The differentiated metric bounds and the
scalar-curvature formula give
\[
|Q_R|\le CR^{-\tau-2},\quad
\|Q_R\|_{L^{n/2}}\le CR^{-\tau},\quad
\|Q_R\|_{L^{2n/(n+2)}}\le CR^{d/2-\tau}.
\]
The Sobolev inequality on the homogeneous completion of
$C_c^\infty(\operatorname{int}N)$ has a constant independent of large $R$
(Euclidean-end Sobolev plus a compact-core Poincar\'e with trace on one annulus,
zero trace at $\Gamma$). Since $\|Q_R\|_{n/2}\to0$, the form
$\mathcal B_R(v,\varphi)=a_n\int\langle dv,d\varphi\rangle_{h_R}+\int Q_Rv\varphi$
is coercive, $\mathcal B_R(v,v)\ge\frac{a_n}2\int|dv|^2$. Solving
$\mathcal B_R(v_R,\varphi)=-\int Q_R\varphi$ and setting $z_R=1+v_R$ gives
\[
(-a_n\Delta_{h_R}+Q_R)z_R=0,\quad z_R|_\Gamma=1,\quad
\|dv_R\|_2+\|v_R\|_{2^*}\le CR^{d/2-\tau}.
\]
Local elliptic regularity makes $v_R$ smooth up to $\Gamma$; on the Euclidean end
it is harmonic and in $L^{2^*}$, so $v_R\to0$ at infinity.
\emph{Uniform and compact convergence.} Cover $\mathcal A_R$ by finitely many balls
of radius comparable to $R$ in $\{R/2<r<4R\}$ and rescale by $R$. The rescaled
metrics are uniformly elliptic with bounded two-jet coefficients, and the rescaled
zeroth-order term $R^2Q_R$ is $O(R^{-\tau})$. The interior supremum estimate gives
$\sup_{\mathcal A_R}|v_R|\le C(R^{-d/2}\|v_R\|_{2^*}+R^2\|Q_R\|_\infty)\le
CR^{-\tau}$. Off the source annulus $v_R$ is harmonic, zero on $\Gamma$, and tends
to zero at the end, so the maximum principle gives
$\|v_R\|_\infty\le CR^{-\tau}$ on all of $N$; in particular $z_R>0$ for large $R$.
On each fixed compact region $\Delta_hv_R=0$ once $R$ is large, and interior and
fixed smooth-boundary estimates give $v_R\to0$ in $C^2$ there. Put
$\widehat h_R=z_R^{4/d}h_R$. The conformal scalar formula gives
\[
R_{\widehat h_R}=z_R^{-(n+2)/d}\bigl(-a_n\Delta_{h_R}z_R+R_{h_R}z_R\bigr)
=z_R^{-4/d}\chi_RR_h\ge0,
\]
and the global bound with $\sup_N|h_R-h|_h=O(R^{-\tau})$ gives the metric
comparison and completeness. The end is $z_R^{4/d}\delta$ with $z_R$ positive
Euclidean harmonic tending to one, hence harmonically flat to every order.
\emph{The ADM limit and the boundary flux.} Write $z_R=1+A_Rr^{-d}+O_j(r^{-d-1})$ on
the Euclidean end, let $\nu$ be the $h$-unit normal at $\Gamma$ pointing into $N$,
and put $I_R=\int_\Gamma\partial_\nu z_R\,dA_h$; boundary $C^2$ convergence gives
$I_R=o(1)$. Integrating $a_n\Delta_{h_R}z_R=Q_Rz_R$ over a large truncation and
letting its radius tend to infinity gives, with the outward normal of the exterior
equal to $-\nu$ at $\Gamma$,
\[
-a_nd\omega A_R-a_nI_R=\int_NQ_Rz_R\,dV_{h_R}.
\]
The nonlinear factor is harmless:
$|\int_NQ_Rv_R\,dV_{h_R}|\le\|Q_R\|_{2n/(n+2)}\|v_R\|_{2^*}\le CR^{d-2\tau}=o(1)$.
For the remaining integral write $b=h-\delta$ and
$\mathcal L(b)=\partial_i\partial_jb_{ij}-\Delta_\delta\tr_\delta b$. On
$\mathcal A_R$, $R_{h_R}\,dV_{h_R}=[\mathcal L(\chi_Rb)+O(R^{-2\tau-2})]\,dx$,
because the nonlinear scalar terms and the volume change are bounded by
$C(|b_R||D^2b_R|+|Db_R|^2)$ with annular integral $O(R^{d-2\tau})$. The outer
linear flux is zero since $b_R=0$ there, and its inner flux is minus the original
ADM flux on $S_R$. Hence
\[
\int_NQ_R\,dV_{h_R}
=-\int_{S_R}(\partial_jh_{ij}-\partial_ih_{jj})n_\delta^i\,dA_\delta
+O(R^{d-2\tau})-\int_{\mathcal A_R}\chi_RR_h\,dV_{h_R}
\longrightarrow-2k\omega m_{\rm ADM}(h).
\]
Since $a_nd=4k$, the flux identity gives $A_R\to\frac12m_{\rm ADM}(h)$ and
$m_{\rm ADM}(\widehat h_R)=2A_R\to m_{\rm ADM}(h)$. Finally
$H_\nu(\Gamma,\widehat h_R)\to H_\nu(\Gamma,h)$ uniformly by compact $C^2$
convergence, so strict negativity persists.
\end{proof}
\subsection{Proof of the main theorem}
For $3\le n\le7$ the frontier is smooth and the assertion is the numerical
Bray--Lee theorem \cite{BrayLee2009}, with the classical positive mass theorem
when the frontier is empty. Assume $n\ge8$. Proposition
\ref{prop:rpi:harmonic-flat-inequality} already proved the nonempty-frontier case
for harmonically flat ends. We now let $(E,h)$ have the general asymptotics of
Theorem~\ref{thm:intro-main} and full perimeter $A>0$.
\emph{The capacitary coefficient at infinity.} Let $w$ be the capacitary
potential and $c$ the capacity. The local minimizing density bounds and the weak
Harnack contraction for the equilibrium potential give its constant extension the
continuous boundary value one on the full frontier. For $0<\alpha<\min\{\tau,1\}$
the function $r^{-d}(1-r^{-\alpha})$ is $h$-superharmonic for all large $r$: its
Euclidean Laplacian is $-\alpha(d+\alpha)r^{-n-\alpha}$, while the metric error is
$O(r^{-n-\tau})$. Comparison with a fixed multiple on a large coordinate sphere
and the maximum principle give $w=O(r^{-d})$, and scaled interior estimates give
$Dw=O(r^{-d-1})$, $D^2w=O(r^{-d-2})$. Consequently
\[
\Delta_\delta w=(\delta^{ij}-h^{ij})\partial_{ij}w
+h^{ij}\Gamma_{ij}^\ell\partial_\ell w=O(r^{-n-\tau}).
\]
Extend a cutoff of $w$, supported in its end chart, by zero to $\R^n$; its
Euclidean Laplacian has a finite $\alpha$-moment. The Newton potential, split into
inner, comparable, and outer annuli, is a monopole plus $O_1(r^{-d-\alpha})$, and
its difference from the extended function is a decaying harmonic function on all
of $\R^n$, hence zero. If the monopole coefficient is $c_0$, flux integration on
$\{w<s\}$ for regular values $s\uparrow1$ gives
$\int_{\{w<s\}}|dw|_h^2\,dV_h=sd\omega c_0$; the flux at infinity is
$-d\omega c_0$ and the other boundary is the regular level, so
$c_0=c$. Thus
\[
0<w<1,\qquad w=cr^{-d}+O_1(r^{-d-\alpha}),\qquad 0<\alpha<\min\{\tau,1\},
\]
and the conformal mass formula applies.
\emph{A strict smooth inner boundary.} Fix small $\lambda>0$ and set
$p=1+\lambda w$, $h_\lambda=p^{4/d}h$. Then $h_\lambda\ge h$, its asymptotically
flat rate is $\min\{\tau,d\}>d/2$, and
$R_{h_\lambda}=p^{-4/d}R_h\ge0$, $R_{h_\lambda}\,dV_{h_\lambda}=p^2R_h\,dV_h$, so
scalar integrability and decay persist. The end expansion gives
\[
m_{\rm ADM}(h_\lambda)=m_{\rm ADM}(h)+2\lambda c .
\]
To apply the static detachment, write this as
$h^{\rm base}_\lambda=(1+\lambda)^{4/d}h$, $\widehat u_\lambda=(1+\lambda w)/(1+\lambda)$,
$h_\lambda=\widehat u_\lambda^{4/d}h^{\rm base}_\lambda$. The normalized harmonic
factor equals one on the frontier and lies strictly between zero and one on the
connected open exterior. Constant scaling preserves local perimeter minimality of
the original filled set, and the smooth ambient metric supplies the minimizing
density and regular-patch estimates. Thus the static detachment conclusion applies
to the original conductor. Proposition~\ref{prop:rpi:one-sided-smoothing} then
gives a smooth compact enclosing boundary $\Gamma$ with
$H_\nu(\Gamma,h_\lambda)<0$. Retain its connected end exterior. The metric
$h_\lambda$ is smooth on a neighborhood of its closure, and all parameters
producing $\Gamma$ are fixed before approximation.
\emph{Comparison of all enclosing areas.} Apply
Lemma~\ref{lem:rpi:smooth-density} on this smooth exterior. Let $k_j$ be the
approximating metrics and choose $\epsilon_j\downarrow0$ so that
$k_j\ge(1-\epsilon_j)h_\lambda$, $H_\nu(\Gamma,k_j)<0$, and
$m_{\rm ADM}(k_j)\to m_{\rm ADM}(h_\lambda)$. Take a minimizing enclosure of the
region removed by $\Gamma$ in $k_j$. Lemma~\ref{lem:g-enclosure} gives compact
existence and Lemma~\ref{lem:g-detachment} detaches the frontier from the strict
smooth obstacle. The resulting frontier $\Sigma_j$ is detached, locally perimeter
minimizing, and outer minimizing, with a connected end exterior, a harmonically
flat end, and scalar curvature zero on that distant end. Every such enclosure
contains the original conductor, so its full perimeter in $h$ is at least $A$.
Tangent-plane metric comparison gives
\[
\mathcal H^{k}_{k_j}(\Sigma_j)\ge(1-\epsilon_j)^{k/2}
\mathcal H^{k}_{h_\lambda}(\Sigma_j)\ge(1-\epsilon_j)^{k/2}A .
\]
Proposition~\ref{prop:rpi:harmonic-flat-inequality} consequently gives
\[
m_{\rm ADM}(k_j)\ge\frac12
\Bigl(\frac{(1-\epsilon_j)^{k/2}A}{\omega}\Bigr)^{d/k}.
\]
First let $j\to\infty$ at fixed $\lambda$, then $\lambda\downarrow0$, using the mass
formula. This gives
\[
m_{\rm ADM}(h)\ge\frac12\Bigl(\frac A\omega\Bigr)^{d/k}.
\]
The strict boundary and elliptic constants may depend on the fixed $\lambda$ and
$\Gamma$; this is consistent with the stated order of limits, and the area
estimate came from metric comparison and conductor inclusion.
\emph{The empty frontier.} A nonempty locally perimeter-minimizing frontier has
positive perimeter by the density lower bound, so $A=0$ means an empty frontier.
On that complete boundaryless manifold apply the same cutoff-density proof with
$\Gamma=\varnothing$; there is then no compact-boundary flux. Applying
Lemma~\ref{lem:rpi:harmonic-pmt} to the harmonic-flat approximants and passing
their masses to the limit gives $m_{\rm ADM}(h)\ge0$. This completes the proof of
Theorem~\ref{thm:intro-main}.
\appendix
\section{A quantitative singular-set bound for weighted minimizers}
\label{sec:rpi:weighted-minimizers}
The smooth positive mass theorem used above is \cite[Corollary~1.6]{BrendleWang2026}.
In its dimension-descent proof, a weighted perimeter minimizer with a volume
forcing is completed across its singular set. This appendix verifies the
quantitative covering estimate that input needs. The conclusion remains the
external theorem; here we only supply the estimate.
The quantitative input is \cite[Theorem~1.3]{NaberValtorta2020}, whose hypotheses
include bounded generalized mean curvature. Bounded mean curvature alone does not
place a singular set in a quantitative stratum of codimension eight, so we use
local minimization to prove that inclusion at every small scale, then apply the
quantitative theorem.
\begin{lemma}[Quantitative estimate for weighted minimizers]
\label{lem:rpi:bw-quantitative}
Let $n\ge8$, let $g$ be smooth on an open set $U$, and let $\widehat\rho>0$ and
$\Phi$ be smooth there. Put $g_*=\widehat\rho^{2/(n-1)}g$ and
$f=\Phi\widehat\rho^{-1/(n-1)}$. Suppose a locally finite-perimeter set $E$
locally minimizes
\[
\mathcal F(E;W)=P_{g_*}(E;W)-\int_{E\cap W}f\,dV_{g_*}
\]
against changes compactly supported in $W\Subset U$. Let $\mathcal S$ be a compact
subset of the singular set of its perimeter support in $U$. Then there are
constants $C,t_0>0$ with
\begin{equation}\label{eq:rpi:bw-tube}
\Vol_g\bigl(B_t^g(\mathcal S)\bigr)\le Ct^8,\qquad 0<t<t_0 .
\end{equation}
The constants are local: they may depend on the minimizer and on a buffered compact
neighborhood of $\mathcal S$.
\end{lemma}
\begin{proof}
The proof has three parts: absorb the density into the metric and get strict
blow-up compactness, exclude highly symmetric cones at every small scale, then
convert that to the tube estimate with the quantitative stratification theorem.
\emph{Absorbing the density; first variation.} The conformal metric absorbs the
positive perimeter density, $d\mathcal H^{n-1}_{g_*}=\widehat\rho\,d\mathcal
H^{n-1}_g$, and $f\,dV_{g_*}=\Phi\widehat\rho\,dV_g$, so the functional is the
weighted perimeter minus weighted volume. Choose
$\mathcal S\subset U_0\Subset U_1\Subset U$. Domain variation of the multiplicity-one
boundary varifold $V=|\partial^*E|$ by a compactly supported field $Y$ gives
\[
\delta_{g_*}V(Y)=\int_{\partial^*E}f\langle Y,\nu_E\rangle_{g_*}\,
d\mathcal H_{g_*}^{n-1},
\]
so $V$ has bounded generalized mean curvature with no extra singular term. The
minimizing comparison gives the almost-minimization estimate
$P_{g_*}(E;W)\le P_{g_*}(F;W)+\|f\|_{L^\infty(U_1)}\Vol_{g_*}(E\triangle F)$ for
$W\Subset U_1$, hence two-sided volume-density and perimeter bounds
$cs^{n-1}\le P_{g_*}(E;B_s(x))\le Cs^{n-1}$ at boundary points.
\emph{Strict blow-up compactness.} For singular points $x_j\in\overline U_0$ and
$r_j\downarrow0$, rescale $g_j=r_j^{-2}\Psi_j^*g_*$, $f_j=r_jf(\Psi_j)$ with
$\Psi_j(y)=\exp_{x_j}(r_jy)$. Then $g_j\to\delta$, $f_j\to0$ smoothly, and
$\chi_{E_j}\to\chi_{E_\infty}$ in $L^1_{\rm loc}$. To upgrade to strict perimeter
convergence, glue $E_\infty$ inside a sphere $B_{s_j}$ to $E_j$ outside, with
$s_j\in(a,b)$ chosen so the trace difference is small; the gluing formula and
minimality give, after letting $b\downarrow a$, strict perimeter convergence on
$B_a$. So the limit is a Euclidean locally perimeter-minimizing boundary with
nonzero two-phase density at zero.
\emph{Excluding highly symmetric cones.} There are uniform $\epsilon,R_0>0$ such
that at every singular point and every scale $0<s<R_0$ the rescaled varifold is not
$\epsilon$-close to an $(n-7)$-symmetric comparison cone. Otherwise a sequence of
comparison cones $C_j$, dilation invariant and translation invariant along
$(n-7)$-planes $K_j\to K$, has a global homogeneous limit $C$ equal to
$|\partial^*E_\infty|$ on $B_1$; homogeneity extends this globally. The radial
direction is tangent, so the phase is constant along radial segments, and $C$
equals the boundary of a conical set $E_0$ that is globally locally perimeter
minimizing. Translation invariance along $K$ makes $E_0=K\times F$ up to a null
set with $F$ a locally perimeter-minimizing conical set in $K^\perp$ of dimension at
most seven, hence regular at its vertex and a halfspace; multiplicity one and the
density bounds make $C$ a hyperplane. Strict convergence and flat-boundary
regularity then contradict singularity at zero. This uses Riemannian regularity with
converging smooth metrics, not a coordinate change of Euclidean mean curvature.
\emph{The tube estimate.} Choose finitely many balls
$B_{\rho_i/2}^{g_*}(p_i)$ covering $\mathcal S$ with $B_{4\rho_i}^{g_*}(p_i)\Subset
U_0$ and $4\rho_i<R_0$, small enough that the rescaled ambient geometry meets the
curvature and injectivity-radius bounds of \cite{NaberValtorta2020}. The first
variation and local perimeter bounds give that theorem's bounded-mean-curvature and
finite-mass hypotheses. Apply it to $B_{2\rho_i}^{g_*}(p_i)$ after rescaling lengths
by $\rho_i^{-1}$; the excluded symmetric cones mean the rescaled singular set lies
in the stratum $S_{\epsilon,r}^{\,n-8}$ for every $0<r<1$, which is stronger than a
statement about tangent cones. The theorem gives rescaled tube volume at most
$C_ir^8$, and returning to physical variables,
\[
\Vol_{g_*}\bigl(B_t^{g_*}(\mathcal S\cap B_{\rho_i/2}^{g_*}(p_i))\bigr)
\le C_i\rho_i^{n-8}t^8 .
\]
Summing over the finite cover and comparing $g_*$ with $g$ on the buffered compact
neighborhood gives \eqref{eq:rpi:bw-tube}. Disjoint equal-radius balls of radius $t$
centered on $\mathcal S$ number at most $Ct^{8-n}$, which is the covering bound used
in \cite[Theorem~3.34]{BrendleWang2026}.
\end{proof}
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