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% Uniformization of complete Kahler manifolds with positive bisectional
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\hypersetup{
pdftitle={Uniformization of complete Kahler manifolds with positive bisectional curvature},
pdfauthor={OpenAI},
pdfsubject={Yau's uniformization conjecture},
pdfkeywords={Kahler manifolds, bisectional curvature, uniformization, Ricci flow, holomorphic discs}
}
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\title{Uniformization of complete K\"ahler manifolds\\[2pt]
with positive bisectional curvature}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
Every complete, connected, noncompact K\"ahler manifold with strictly
positive holomorphic bisectional curvature is biholomorphic to complex
Euclidean space. This settles Yau's uniformization conjecture in every
complex dimension, with no additional hypotheses on volume growth,
curvature bounds, or topology.
\end{abstract}
\section*{What this paper proves, in plain words}
A K\"ahler manifold is a space that looks locally like $\C^n$ and carries
a metric whose imaginary part is a closed two-form. Positive holomorphic
bisectional curvature says the space bends like a round sphere in every
complex two-plane. The theorem says: if such a space has no boundary and
no holes in the metric sense (it is complete and noncompact), then as a
complex manifold it is exactly $\C^n$.
The proof runs K\"ahler--Ricci flow $h_t$ (an equation that shrinks the
metric in the direction of its Ricci curvature) and shows three things:
the flow exists forever, it carries enough curvature information to
produce explicit shrinking holomorphic coordinates near every point, and
those coordinates can be glued into one global biholomorphism onto $\C^n$.
\section*{How to read this paper}
Three reading paths:
\begin{itemize}
\item \emph{One hour.} Read this page, the notation table, the road map,
Section~\ref{sec:introduction}~(\S\ref{sec:structure}), and the
statements of Theorem~\ref{thm:uniformization},
Theorem~\ref{thm:localized-flow},
Theorem~\ref{thm:joint-positivity},
Proposition~\ref{prop:shrinking-charts},
Proposition~\ref{prop:polynomial-models},
Proposition~\ref{prop:global-coordinates}.
\item \emph{One day.} Add every theorem, proposition, lemma, and
corollary statement, in order. Every proof can be skipped on this path.
\item \emph{Checking the argument.} Every definition, statement and proof
in the paper. All numerical claims are stated; no hidden inputs.
\end{itemize}
The three time variables play distinct roles and are easy to confuse:
\[
\begin{array}{l|l|l}
\text{clock} & \text{definition} & \text{role}\\\hline
\text{physical} & t & \text{K\"ahler--Ricci flow}\\
\text{logarithmic} & \tau=\log t & \text{joint positivity, Harnack}\\
\text{volume} & s=\rho(o,t) & \text{chart contraction, degree bounds}
\end{array}
\]
In Sections~\ref{sec:discs}--\ref{sec:variation} the letter $s$ is
temporarily reused for the physical time at a disc center. From
Section~\ref{sec:charts} onward it always means the volume clock and
nothing else.
\section*{Notation}
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Symbol & Meaning \\\midrule
$M$ & connected noncompact complex manifold, $\dim_\C M=n$ \\
$J$ & complex structure \\
$g$ & complete K\"ahler metric, initial data \\
$h_t$ & K\"ahler--Ricci flow $h_t$ with $h_0=g$ \\
$\omega_g$ & K\"ahler form; $\omega_h$ for $h_t$ \\
$R_g$, $\Ric(g)$, $\Rm$ & scalar, Ricci, Riemann curvatures \\
$\dd f$ & complex Hessian $(f_{i\bar j})$ \\
$\Delta_h f$ & $\tr_h\dd f$\\
$b_t = h_t^{-\mathsf T}$ & cometric in matrix form on the cotangent bundle \\
$N(x,\xi,t)$ & $\xi^* b_t(x)\,\xi$ on $T^{*(1,0)}M$ \\
$\rho$ & $\log(\det g/\det h_t)$, the volume-loss potential \\
$\tau=\log t$ & logarithmic time \\
$s=\rho(o,t)$ & volume clock at base point $o$ \\
$q(s)$ & $ds/d\tau = tR(o,t)$\\
$\Phi_s$, $f_s$ & unitary chart at clock $s$, its continued inverse \\
$F_{s,u}$ & transition germ $\Phi_u^{-1}\Phi_s$\\
$G_k$, $H_k$ & polynomial model, holomorphic coordinate change \\
$v$ & disc envelope of $N$ on $T^{*(1,0)}M \times (0,T)$\\
$T$ & horizon in the disc and envelope arguments\\\bottomrule
\end{tabular}
\end{center}
\section*{Road map}
\begin{center}
\begin{tabular}{@{}lp{0.36\textwidth}p{0.36\textwidth}@{}}
\toprule
Sec. & What happens & Why it is needed\\\midrule
\ref{sec:introduction} & statement, context, structure & orientation \\
\ref{sec:preliminaries} & global functions, sections, exhaustions & input data for the flow \\
\ref{sec:flow} & localized K\"ahler--Ricci flow, scalar comparison & lower bound for $\rho$ \\
\ref{sec:discs}--\ref{sec:ellipsoids} & holomorphic discs, ellipsoids & joint positivity in space and time \\
\ref{sec:charts} & positive-time completeness, Harnack, simple connectivity, shrinking charts & exponential contraction of one exponent \\
\ref{sec:dynamics} & polynomial models of chart transitions & normalized coordinates under nonuniform contraction \\
\ref{sec:uniformization} & backward iteration, surjectivity & global biholomorphism onto $\C^n$\\
\bottomrule
\end{tabular}
\end{center}
\tableofcontents
% =============================================================================
\section{Introduction}\label{sec:introduction}
In his 1982 survey, Yau asked whether a complete noncompact K\"ahler
manifold with positive holomorphic bisectional curvature must be
biholomorphic to complex Euclidean space
\cite[\S9(b), p.~45]{YauProblems}. We prove this, with no additional
hypotheses.
\begin{theorem}\label{thm:uniformization}
Let $M$ be a connected noncompact complex manifold of complex dimension
$n\ge 1$. If $M$ carries a smooth complete K\"ahler metric with strictly
positive holomorphic bisectional curvature, then $M$ is biholomorphic
to $\C^n$.
\end{theorem}
Strict positivity is pointwise: no uniform lower bound, no upper
curvature bound, no volume-growth, noncollapsing, Ricci-pinching, or
topological hypothesis is added. The conclusion is biholomorphism onto
\emph{all} of $\C^n$, not onto a proper domain, and it does not claim
the given metric is Euclidean. In particular, $M$ is Stein and
contractible.
\subsection{Antecedents and the remaining analytic difficulties}
\label{sec:antecedents}
In one complex dimension, the result is classical: a complete
noncompact Riemann surface with positive Gaussian curvature is
conformally the plane, by the Cohn--Vossen topology theorem and the
Blanc--Fiala--Huber parabolicity theorem
\cite[Satz~8]{CohnVossen}\cite[Theorem~15]{Huber}.
In the compact case, the Frankel conjecture was proved by Mori and by
Siu--Yau: positive holomorphic bisectional curvature characterizes
complex projective space \cite{MoriAmple,SiuYauFrankel}. Yau's
noncompact question \cite[\S9(b), p.~45]{YauProblems} asks for the
analogous characterization of $\C^n$, allowing arbitrary geometry at
infinity.
\emph{Approach via functions and embeddings.}
Greene--Wu showed that positive real sectional curvature yields a smooth
strictly convex exhaustion, hence Steinness
\cite[Theorem~10]{GreeneWuConvex1976}. Mok embedded the manifold into
an affine-algebraic variety under positive bisectional curvature,
maximal volume growth, and quadratic scalar-curvature decay
\cite{MokEmbedding1984}. Chen--Zhu proved intrinsic growth and
average-curvature estimates under pointwise positive bisectional
curvature, with a linear average-scalar-curvature bound whose constant
may depend on the center \cite{ChenZhu}; they also studied algebraic
compactification under bounded sectional curvature and a finite top
Ricci integral \cite{ChenZhuPositive2009}. Ni--Tam studied heat
deformation of plurisubharmonic functions under nonnegative bisectional
curvature, keeping an explicit growth condition on the initial
function \cite{NiTamStructure}.
\emph{Approach via geometric flow.}
Cao's differential Harnack inequalities for K\"ahler--Ricci flow
\cite{CaoHarnack} underlie everything below. Chau--Tam established
uniformization under bounded nonnegative bisectional curvature and
maximal volume growth \cite{ChauTamComplex2006}, and later developed
locally biholomorphic charts, covering and pseudoconvexity, and
soliton-limit methods under further decay or pinching assumptions
\cite{ChauTamStein,ChauTamSimply}. The difference between a covering,
a map onto a domain, and a biholomorphism onto $\C^n$ matters here.
Liu proved uniformization under nonnegative bisectional curvature and
maximal volume growth, without upper curvature bound, via a
Gromov--Hausdorff argument that retracts holomorphic vector fields
\cite{LiuUniformization}; Lee--Tam later obtained the same conclusion
by smoothing the metric and applying the bounded-curvature theorem
\cite{LeeTam}. Both retain maximal volume growth.
\emph{Current frontier.}
Recent work removes assumptions at infinity in special cases.\footnote{%
All cited versions record AI assistance. Datar--Pingali--Seshadri credit
ChatGPT Pro~5.2 for the normalization observation in Lemma~5.13
\cite[\S1.1, Acknowledgments]{DatarPingaliSeshadri}, and ChatGPT~5.5~Pro
for the auxiliary function and induction in the B\'ezout estimates,
plus ChatGPT~5.0 in thinking mode for Lemma~2.2
\cite[\S\S1--3]{DatarPingaliSeshadriTop}. Wu reports ChatGPT-6
assistance, with author verification
\cite[Acknowledgments]{WuConditional}.}
Datar--Pingali--Seshadri \cite[Theorem~1]{DatarPingaliSeshadri} prove
that every complete noncompact K\"ahler surface with positive real
sectional curvature is biholomorphic to $\C^2$, via Lipschitz
plurisubharmonic weights of finite Monge--Amp\`ere mass and weighted
holomorphic functions; their Theorem~2 gives the same for positive
bisectional curvature under strong Steinness, contractibility and
simple connectivity at infinity. Wu
\cite[Corollaries~1.5--1.6]{WuConditional} removes contractibility,
retaining strong Steinness and simple connectivity at infinity; he also
handles positive bisectional curvature when real sectional curvatures
are nonnegative outside a compact set. These are surface results, and
positive real sectional curvature is stronger than positive bisectional
curvature.
In higher dimensions, Datar--Pingali--Seshadri
\cite[Theorem~1.1]{DatarPingaliSeshadriTop} prove finiteness of the
integral of the top power of the Ricci form on complete noncompact
K\"ahler manifolds with positive real sectional curvature, and the same
under positive bisectional curvature given a smooth strictly
plurisubharmonic exhaustion with uniformly bounded gradient, with
quasiprojectivity under bounded positive sectional curvature
\cite[Theorem~1.2]{DatarPingaliSeshadriTop}. Dat \cite{DatHessian}
develops an $m$-Hessian capacity approach in complex dimension $\ge3$;
his uniformization route retains proper-map and multiplicity
assumptions. Shi \cite[Theorem~1.6]{ShiMinimalDegrees} proves an
identity between the minimal degree of holomorphic volume forms and
refined minimal degrees of holomorphic functions for complete
noncompact K\"ahler manifolds with nonnegative bisectional curvature and
nonsplitting universal cover, tying the function-theoretic and flow
pictures together in the maximal-growth regime
\cite[Theorem~1.7, Corollary~1.12]{ShiMinimalDegrees}.
\emph{Two obstructions in the unrestricted problem.}
First, the initial metric can have unbounded curvature and arbitrarily
collapsed regions, so a bounded-curvature flow theorem or a global
tensor maximum principle cannot be invoked before its hypotheses are
established. Second, shrinking charts alone identify $M$ with a proper
domain of $\C^n$; surjectivity requires quantitative control of the
chart changes even when contraction rates vary substantially.
\subsection{Structure of the proof}\label{sec:structure}
Fix $o\in M$. We construct a global K\"ahler--Ricci flow $h_t$ by
complete Hermitian approximations on exhausting domains. Estimates for
$h_t$ and for the volume-loss potential
$\rho(x,t)=\log(\det g(x)/\det h_t(x))$ pass to the limit without a
global curvature bound. A spatial weight adapted to the nonnegative
form $g-h_t$ proves scalar comparison before completeness of $h_t$ is
known. Order matters here: completeness of positive-time slices is a
\emph{consequence} of the Harnack inequality, not an assumption.
The curvature argument is on the holomorphic cotangent bundle. Minimize
the boundary squared dual norm over holomorphic discs, with compact
boundary walls and a vanishing area penalty. The canonical symplectic
bivector relates the complex Hessian of the norm to its time derivative.
A quantitative deformation lemma gives a viscosity inequality for the
disc infimum; optimizing a Hermitian ellipsoid in each fiber converts it
into the scalar comparison already proved. The infimum therefore equals
the evolving squared dual norm, and $N(x,\xi,e^{w+\bar w})$ is jointly
plurisubharmonic in space and logarithmic time. That yields the scalar
and matrix Harnack inequalities.
The volume clock $s=\rho(o,t)$ measures contraction directly. For a
transition $F_{s,u}$ between centered unitary charts at clocks $s$ and
$u\ge s$,
\[
\abs{\det F_{s,u}'(0)}=e^{-(u-s)/2}.
\]
Writing $R(x,t)$ for the scalar curvature of $h_t$, the speed of the
volume clock in logarithmic time is $q(s)=ds/d\tau=tR(o,t)$. A static
logarithmic-norm estimate bounds the greatest singular length of an
$h_t$-unitary local chart, measured in the initial metric $g$, by a
fixed power of its least singular length; together with the
determinant loss, this controls contraction in every direction.
Selected time intervals with Ricci pinching then give a loop
contraction argument. Simple connectivity lets the local inverse charts
continue coherently over exhausting compact sets, and a
plurisubharmonic Liouville argument gives one contraction exponent
valid on every fixed compact set.
Finally, normalize the volume jets of the transition maps and build
polynomial automorphisms of constant Jacobian. The nonlinear shears used
to realize normalized jets have determinant one, so the full models
retain a contracting linear determinant. Intervals of nonuniform
contraction are permitted; their degree cost is charged to growth of
$q$ in the volume clock. Backward iteration of these models makes the
corrected inverse charts converge to an injective holomorphic map
$M\to\C^n$. A subsequential degree bound for the forward compositions,
together with smallness on compact subsets of the image and an
inverse-ball inclusion, eliminates a finite maximal radius for centered
balls in the image. Hence the limiting coordinate map is onto $\C^n$.
Sections~\ref{sec:preliminaries}--\ref{sec:flow} build the analytic
data and prove scalar comparison. Sections~\ref{sec:discs}--\ref{sec:ellipsoids}
prove joint positivity via discs and ellipsoids. Section~\ref{sec:charts}
obtains completeness, simple connectivity and quantitative shrinking
charts. Sections~\ref{sec:dynamics}--\ref{sec:uniformization} construct
the biholomorphism. The disc-deformation estimate and the forward-degree
estimate are stated with exactly the uniformities their later
applications need.
\subsection{Conventions}\label{sec:conventions}
We use the real curvature convention in which $\Rm(a,b,b,a)$ is the
sectional-curvature numerator of the real plane spanned by $a,b$. The
hypothesis is
\[
\Rm(u,v,v,u)+\Rm(u,Jv,Jv,u)>0
\]
for every pair of real unit tangent vectors at every point. By the
K\"ahler symmetries this is equivalent to strict Griffiths positivity of
$T^{1,0}M$ in complex notation. In complex dimension one the displayed
sum is the Gaussian curvature.
We identify a Hermitian tensor $(a_{i\bar j})$ with its real
$(1,1)$-form, dropping the factor $\sqrt{-1}$ when no confusion can
result. Thus $\dd f$ is the complex Hessian, $\Delta_h f=\tr_h\dd f$,
and
\[
\partial_t h_t=-\Ric(h_t),\qquad
\Ric(h)=-\dd\log\det h.
\]
The dual of a Hermitian metric includes the transpose dictated by the
holomorphic cotangent pairing; matrix contractions retain it explicitly
where needed.
The initial complete metric is $g$, the base point is $o$, and
$r(x)=d_g(o,x)$. Constants may change between estimates; subscripts
record dependencies when those matter. For Hermitian forms,
$A=O(b)h$ means $-Cbh\le A\le Cbh$. The abbreviation psh means
plurisubharmonic. A positive constant extracted from curvature on a
specified compact set is never uniform over all of $M$.
% ---------------------------------------------------------------------------
% End of front matter and introduction. Next sections appended below.
% =============================================================================
\section{Functions, sections, and smoothing on the initial manifold}
\label{sec:preliminaries}
Throughout this section, distances, balls, norms, and volumes are those
of the complete initial metric $g$. Fix $o\in M$ and put $r(x)=\dist_g(o,x)$.
Constants may depend on $g$ and $o$, but not uniformly on the center.
We need four pieces of analytic data before we can run a flow or study
holomorphic discs. First, a strictly plurisubharmonic exhaustion with
controlled gradient and complex Hessian; second, finitely many global
holomorphic functions and vector fields that detect every tangent
direction; third, global holomorphic canonical sections with linear
logarithmic growth; fourth, a smoothing procedure that produces a
two-sided complex Hessian bound on distance truncations even when no
curvature upper bound is available. Those make up
Proposition~\ref{prop:initial-data} and the lemmas of the two
subsections below.
\begin{proposition}[Initial analytic data]\label{prop:initial-data}
The following objects and estimates exist.
\begin{enumerate}[label=\textup{(\roman*)}]
\item A smooth strictly plurisubharmonic $\psi$ with $\psi\le C(1+r)$.
\item Holomorphic functions and holomorphic tangent fields that
interpolate arbitrary finite jets at a finite set of points.
In particular, finitely many holomorphic functions define an
injective immersion $F:M\to\C^N$, and finitely many global
holomorphic tangent fields span $T^{1,0}M$ pointwise.
\item At every $x\in M$ there is a global holomorphic canonical section
$S$ with $S(x)\ne0$ and $\log|S|_g^2\le C_S(1+r)$. There is also a
smooth $\Xi$ with
\begin{equation}\label{prelim:xi}
\dd\Xi\ge\Ric(g),\qquad \Xi\le C(1+r^{3/2}).
\end{equation}
\item A smooth exhaustion $u\ge1$ satisfying
\begin{equation}\label{prelim:exhaustion}
C^{-1}(1+r)\le u\le C(1+r),\qquad
|\partial u|_g\le C,\qquad |\dd u|_g\le C/u.
\end{equation}
\item If $S(o)\ne0$ is chosen as in \textup{(iii)}, then for $L\ge1$,
\begin{equation}\label{prelim:averages}
\frac1{\Vol B(o,L)}\int_{B(o,L)}\bigl|\log|S|_g^2\bigr|\,dV_g
\le CL,\qquad
\frac1{\Vol B(o,L)}\int_{B(o,L)}R_g\,dV_g\le C/L.
\end{equation}
\end{enumerate}
Neither $F$ nor $\psi$ is proper. The exhaustion in \textup{(iv)} has a
two-sided complex Hessian bound, which is what the flow arguments
actually use.
\end{proposition}
The rest of this section proves the proposition. Subsection~\ref{sec:weight}
constructs $\psi$ (via a Busemann function) and the holomorphic
interpolation data. Subsection~\ref{sec:hodge} proves the Hodge-heat
order and commutation facts needed later. Subsection~\ref{sec:dyadic}
combines both to obtain the dyadic exhaustion and the averaged bounds.
\subsection{The weight and holomorphic interpolation}\label{sec:weight}
Let $\gamma$ be a unit-speed ray starting at $o$ and set
\[
b(x)=\lim_{j\to\infty}\bigl\{d_g(o,\gamma(j))-d_g(x,\gamma(j))\bigr\}.
\]
The triangle inequality gives monotonicity in braces, uniform boundedness
on compact sets, and $|b(x)-b(y)|\le d_g(x,y)$.
\paragraph{Busemann is strictly plurisubharmonic.}
Fix $x$ in a compact set and a minimizing segment of length $l$ from $x$
to $\gamma(j)$. For an initial vector $X$, use its parallel translate
multiplied by $1-\sigma/l$ as a comparison variation field, and the same
construction for $JX$. The second variation of length, computed on
their perpendicular components, has curvature contribution the negative
of
\[
\int_0^l\Bigl(1-\tfrac{\sigma}{l}\Bigr)^2
\bigl\{\Rm(\dot\gamma,X_\parallel,X_\parallel,\dot\gamma)
+\Rm(\dot\gamma,JX_\parallel,JX_\parallel,\dot\gamma)\bigr\}
\,d\sigma,
\]
which is a positive multiple of the bisectional curvature of the complex
two-plane spanned by $\dot\gamma$ and $X$. Removing radial components
does not change the expression. On a fixed short initial subsegment,
positivity is bounded below uniformly in the unit initial vector
$X$ and the base point in the chosen compact. The derivative term in
the index form is $O(l^{-1})$. Hence
\[
\partial^2_{X,\bar X}\!d_g(\cdot,\gamma(j))
+ \partial^2_{JX,\overline{JX}}\!d_g(\cdot,\gamma(j))
\le -c_K|X|^2 + O(l^{-1})|X|^2
\]
with $c_K>0$ independent of $j$.
This comparison does not need a smooth distance function. Vary the
initial endpoint along a holomorphic germ and use the comparison paths
to obtain smooth upper supports for distance. The sum of the endpoint
acceleration terms vanishes for the two real directions of that germ,
so their negatives give lower supports for the signed distance with the
same strict Levi lower bound. An upper test of the signed distance also
lies above its lower support and therefore inherits that bound. Local
uniform passage to $b$ preserves the inequality. Testing on complex
lines gives the subharmonic version. Hence $b$ is strictly
plurisubharmonic, with a positive Levi lower bound on every compact.
Richberg approximation with pointwise error less than one gives a smooth
strictly plurisubharmonic $\psi$ with $\psi\le1+r$, after adding a
constant \cite{Richberg}\cite[Section~4, Corollary~2]{GreeneWu}.
\paragraph{Holomorphic interpolation.}
Here are the bundle choices. Let $K=\bigwedge^nT^{*(1,0)}M$. A function,
a canonical section, and a tangent field correspond respectively to a
holomorphic top form with coefficients in
\begin{equation}\label{prelim:coefficient-bundles}
K^{-1},\qquad \boldone,\qquad T^{1,0}M\otimes K^{-1}.
\end{equation}
The first has nonnegative curvature; the second is flat; the third has
nonnegative Nakano curvature. Indeed it is $E\otimes\det E$ with
$E=T^{1,0}M$ Griffiths-positive, so the Demailly--Skoda
tensor-with-determinant theorem applies \cite[Theorem~2.6]{Demailly}.
These statements concern the bundles in \eqref{prelim:coefficient-bundles};
no Nakano positivity of $E$ itself is being assumed.
Prescribe jets of order $m$ at a finite set $A$. Choose disjoint
coordinate balls and local holomorphic sections with those jets,
multiply them by cutoffs equal to one on smaller balls, obtaining a
smooth section $s_0$ with $\dbar s_0$ supported in a compact set
disjoint from $A$. Use the weight
\[
k\psi+\sum_{a\in A}c_m\vartheta_a\log|z_a|^2,\qquad c_m>n+m,
\]
where each $\vartheta_a$ is one near $a$ and supported in its coordinate
ball. The negative Hessian terms of the cutoff logarithms are supported
in a fixed compact, and a sufficiently large $k$ makes the total weight
strictly plurisubharmonic and dominant there. The inverse-curvature norm
of $\dbar s_0$ is integrable because its support avoids the poles. The
complete K\"ahler $L^2$ estimate for bundle-valued top forms gives
$v$ with $\dbar v=\dbar s_0$ and finite weighted norm
\cite{Hormander}\cite[Theorem~5.1]{Demailly}. The singular-weight
statement follows by regularizing the logarithms and taking a weak
limit. Near a point of $A$, $v$ is holomorphic, and integrability against
$|z_a|^{-2c_m}$ forces its Taylor coefficients through order $m$ to
vanish. Hence $s_0-v$ has the prescribed jets. Applying this to pairs of
points proves point separation.
For a canonical section the same construction yields
\[
\int_M |S|_g^2e^{-k\psi}\,dV_g<\infty.
\]
The cutoff pole weights can be dropped at the cost of a constant.
$|S|_g^2$ is subharmonic: away from its zeros,
$\dd\log|S|_g^2=\Ric(g)$, and across zeros its squared norm still has
nonnegative distributional Laplacian. The local mean-value inequality on
a unit ball \cite[Theorem~2.1]{LiSchoen}, combined with the upper bound
for $\psi$, gives
\[
|S(x)|_g^2\le\frac{C_S}{\Vol B(x,1)}\exp(C_S(1+r(x))).
\]
Nonnegative Ricci curvature gives
$\Vol B(x,1)\ge c(1+r(x))^{-2n}$ via Bishop--Gromov relative volume
comparison \cite[\S2.1]{GromovBetti}. The polynomial loss is absorbed by
the exponential. This proves \textup{(iii)} and the growth assertion,
matching the canonical-section construction in
\cite[Section~3]{ChenZhu}.
\paragraph{Finite tuples.}
Why finitely many functions and fields suffice is a dimension count, not
an approximation argument. The compact-open spaces of holomorphic
functions and sections are Fr\'echet, hence Baire. Fix a compact family
of points, or of distinct pairs of points. Surjectivity of
finite-jet evaluation, openness of matrix rank, and a finite cover give
a finite-dimensional space of perturbations on which the evaluation
derivative is onto on a neighborhood of that compact family; the
derivative is independent of the perturbation parameter, because
evaluations are affine. Therefore the universal evaluation is a
submersion there.
For each smooth stratum of the forbidden locus, its inverse image under
the evaluation has real codimension equal to that of the stratum. If
that codimension exceeds the real dimension of the point or pair space,
the inverse image has dimension less than the perturbation space. Its
projection has measure zero: cover by countably many compact coordinate
patches, use Lipschitz projection on each, and cover a $d$-dimensional
cube by $O(\delta^{-d})$ small cubes to see the image in $\R^p$, $p>d$,
has volume at most $C\delta^{p-d}$.
For $N$ functions the rank-$r$ stratum of $N\times n$ derivative
matrices has complex codimension $(N-r)(n-r)\ge N-n+1$ when $r<n$.
For $N\ge2n+1$ this exceeds $n$. The equal-values condition on distinct
pairs has complex codimension $N>2n$. Avoidance on a compact set is
open in the compact-open topology via Cauchy estimates. A countable
exhaustion of $M$ and of $M\times M\setminus\operatorname{diag}$, then
Baire, gives one finite injective immersion. For $m$ tangent sections
the least deficient-rank codimension is $m-n+1>n$ when $m\ge2n$; the
same zeroth-jet argument gives a finite spanning family. Neither is
claimed proper.
\paragraph{The function $\Xi$ in \textup{(iii)}.}
Choose countably many canonical sections $S_i$ with no common zero.
Write $a_i=|S_i|_g^2$; the bound $a_i\le\exp(C_i(1+r))$ implies
\[
a_i\le e^{B_i}\exp(1+r^{3/2}).
\]
Choose $c_i>0$ so small that $c_ie^{B_i}\le2^{-i}$ and every derivative
of $c_ia_i$ of order at most $i$ has norm at most $2^{-i}$ on the first
$i$ exhaustion compacts. The sum $\sum c_ia_i$ converges smoothly
locally, is positive, and is bounded above by $\exp(1+r^{3/2})$. In a
local canonical frame it is $\det(g)^{-1}$ times a sum of squares of
holomorphic functions. Hence $\Xi=\log\sum_i c_i|S_i|_g^2$ satisfies
\eqref{prelim:xi}.
\subsection{Hodge heat without a curvature upper bound}\label{sec:hodge}
To smooth distance truncations later, we need to preserve an upper
bound on their complex Hessian. Scalar heat estimates then control the
trace and give the missing lower Hessian bound. We record here how heat
evolution of a bounded closed $(1,1)$-form produces a scalar potential
for its change. That identity will be used again when constructing the
flow comparison cutoff.
Let $H_a$ be scalar heat for $\Delta_g$. Normalize the Hodge heat
operator $\mathcal H_a$ on real $(1,1)$-forms to agree with $H_a$
after taking trace, and normalize so that its generator on closed real
$(1,1)$-forms is $A\mapsto \dd\tr_g A$.
\paragraph{Mass of the scalar heat kernel.}
Fix a pole $y$ and let $\chi_R$ be a Lipschitz distance cutoff equal to
one on $B_g(y,R)$, supported in $B_g(y,2R)$, with $|\nabla\chi_R|\le
C/R$. The Gaussian estimate of Li--Yau \cite{LiYau} and the kernel
gradient estimate of Souplet--Zhang \cite[Theorem~1.3]{SoupletZhang},
together with Bishop--Gromov comparison \cite[\S2.1]{GromovBetti}, give
\[
\int_{R\le d(x,y)\le2R}|\nabla_xH_s(x,y)|\,dV_g(x)
\le Cs^{-1/2}(1+R/\sqrt s)^N e^{-cR^2/s}
\]
for a fixed dimensional exponent $N$. Integrating the heat equation
against $\chi_R$ from $\varepsilon$ to $t\le T$ and letting
$\varepsilon\downarrow0$ yields
\[
\Bigl|\int_M\chi_R(x)H_t(x,y)\,dV_g(x)-1\Bigr|
\le\frac CR\int_0^T s^{-1/2}
(1+R/\sqrt s)^N e^{-cR^2/s}\,ds,
\]
and the right side tends to zero as $R\to\infty$. Positivity and the
sub-Markov property of the minimal kernel allow passage to the total
mass, so $H_t1=1$. The cutoffs only need completeness and the first
derivative of distance; they precede the smooth exhaustion below.
\begin{lemma}[Heat order and commutation]\label{lem:hodge-heat}
For bounded real $(1,1)$-forms, Hodge heat is well defined by cutoff
limits of the complete $L^2$ semigroup. It preserves order and
\[
-Cg\le A\le Cg\ \Longrightarrow\
-Cg\le\mathcal H_aA\le Cg,\qquad
\tr_g\mathcal H_aA=H_a\tr_gA.
\]
If $A$ is also smooth and closed, then
\begin{equation}\label{prelim:hodge-identity}
\mathcal H_aA=A+\dd\int_0^aH_b\tr_gA\,db.
\end{equation}
If a Lipschitz $f$ is constant outside a compact set and $\dd f\le Cg$
as currents, then
\begin{equation}\label{prelim:hessian-preservation}
\dd H_af\le Cg\qquad(a>0).
\end{equation}
\end{lemma}
\begin{proof}
For smooth compactly supported data, take the componentwise
zero-Dirichlet Hodge heat on a relatively compact smooth domain. Its
Weitzenb\"ock formula is a connection Laplacian plus a zeroth-order
curvature reaction. At a null vector $X$ of a nonnegative Hermitian
form $A$, diagonalize $A$ in a unitary frame; the reaction on
$(X,\bar X)$ is a positive fixed multiple of
\[
\sum_j R_g(X,\bar X,e_j,\bar e_j)A_{j\bar j}\ge0
\]
and the Ricci-times-$A$ terms vanish there. The tensor maximum principle
preserves the nonnegative cone. Apply the same argument to $Cg-A$ to
get the upper bound. The trace solves the scalar heat equation with
zero boundary condition. This is the null-eigenvector mechanism of
\cite[Section~2]{NiTamStructure}.
\emph{Complete realization and exhaustion.} On compactly supported
smooth forms put $D=d+d^*$. Completeness gives essential self-adjointness
of $D^2$, hence of $D$: a deficiency vector for $D$ would lie in
$\ker((D^2)^*+1)=0$
\cite[Corollary~2.10, Remark~2.12]{BravermanMilatovicShubin}. The
quadratic form of the resulting Hodge Laplacian is
\[
q(A)=\|dA\|_2^2+\|d^*A\|_2^2,\qquad
V=\overline{C^\infty_c}^{\,(\|\cdot\|_2^2+q)^{1/2}}.
\]
For a nested smooth exhaustion $\Omega_j$, let $V_j$ be the same
closure of forms supported in $\Omega_j$, extended by zero. On a fixed
domain, local ellipticity identifies this with the full zero
boundary-value section domain (not an absolute or relative Hodge
condition). The $V_j$ are nested and their union is dense in $V$. For
$\lambda>0$ and an $L^2$ form, the domain resolvent applied to it and
extended by zero is the orthogonal projection of the complete resolvent
onto $V_j$ for the inner product $q+\lambda\langle\cdot,\cdot\rangle$:
both satisfy the same variational identity against $V_j$. Hence the
projections converge in that norm. Strong resolvent convergence and the
common lower bound zero give strong heat convergence
\cite[Section~6.6, Theorem~6.31, Corollary~6.33]{Teschl2009}. Here
$\Delta_j$ denotes the nonnegative domain Hodge Laplacian; zero
extension of its heat means $e^{-a\Delta_j}P_j$, with $P_j$ restriction
followed by zero-extension of the generator.
\emph{Cone and trace passes to the limit.} The complete Hodge semigroup
has the off-diagonal estimate
\begin{equation}\label{prelim:off-diagonal}
\|\boldone_E\mathcal H_a\boldone_F\|_{2\to2}
\le C\exp(-c\dist(E,F)^2/a).
\end{equation}
This needs no curvature upper bound: use the self-adjoint first-order
operator $d+d^*$, whose principal symbol has norm equal to the covector
norm, so the wave evolution has finite propagation speed
\cite[Theorem~1.1, Section~4]{McIntoshMorris}. The Gaussian Fourier
representation of $\exp(-a(d+d^*)^2)$ then gives
\eqref{prelim:off-diagonal}, up to the constants in our normalization
\cite[Theorem~3.4]{CoulhonSikora}. Since $\Vol B(o,R)\le CR^{2n}$,
dyadic annular decomposition makes the cutoffs of a bounded form
Cauchy in $L^2$ on each compact, uniformly for bounded positive times.
Interior parabolic estimates give smooth convergence away from $a=0$.
\emph{Closedness.} Use the complete semigroup, not the Dirichlet
operators on domains. Let $\zeta_R$ be a compactly supported Lipschitz
cutoff equal to one on $B(o,R)$, supported on $B(o,2R)$ with
$|d\zeta_R|\le C/R$. The complete Hodge semigroup commutes with $d$ on
the $L^2$ domains, by the self-adjoint de Rham complex, and
$De^{-aD^2}=e^{-aD^2}D$ gives the $d$ identity in degree one higher.
The data $\zeta_RA$ belong to the full $D$ graph domain by compactly
supported $H^1$ approximation. For bounded closed $A$,
\[
d\mathcal H_a(\zeta_RA)=\mathcal H_a^{(3)}(d\zeta_R\wedge A).
\]
The data on the right have $L^2$ norm at most $CR^{n-1}$ and support
outside $B(o,R)$; \eqref{prelim:off-diagonal} in degree three makes them
tend to zero on each compact, uniformly for $0<a\le a_0$. Distributional
passage to the limit gives $d\mathcal H_aA=0$, and local regularity
gives its initial value $A$. K\"ahler commutation on closed forms now
gives
\[
\partial_a\mathcal H_aA=\dd\tr_g\mathcal H_aA=\dd H_a\tr_gA.
\]
Integration proves \eqref{prelim:hodge-identity}, first distributionally
then as smooth forms for $a>0$. The same passage from closed-form heat
to a scalar potential via its trace is the mechanism in
\cite[Lemma~2.1, proof of Theorem~2.1]{NiTamHodge2013}.
\emph{Hessian preservation.} Put $f_0=f-c$ with $c$ the value at
infinity; this is compactly supported and Lipschitz. Complete $L^2$
K\"ahler commutation, smooth approximation, and duality give, for a
compactly supported nonnegative Hermitian test form $B$,
\[
\langle\dd H_af_0,B\rangle=\langle\dd f_0,\mathcal H_aB\rangle.
\]
The metric pairing identifies the cone of nonnegative Hermitian forms
with its dual; equivalently one pairs with the Hodge star. Choose
$0\le\zeta\le1$ compactly supported and equal to one near
$\supp f_0$. Since $\dd f_0$ is supported there,
\[
\langle\dd f_0,\mathcal H_aB\rangle
=\langle\dd f_0,\zeta\mathcal H_aB\rangle
\le C\int\zeta\tr_g\mathcal H_aB\,dV_g
\le C\int\tr_gB\,dV_g.
\]
The last two steps use positivity and scalar trace commutation. Since
$H_a1=1$ under nonnegative Ricci curvature, $\dd H_af=\dd H_af_0$.
\end{proof}
\subsection{Dyadic smoothing and averaged curvature}\label{sec:dyadic}
For $L=2^j$, $j\ge0$, choose a smooth nondecreasing scalar truncation of
$r^2/L^2$ which vanishes for $r\le L$ and equals one for $r\ge2L$, and
call it $f_L$. It is Lipschitz with constant $C/L$. The paired
index-form comparison with linear parallel fields gives
$\dd r^2\le Cg$ away from the cut locus, and in the upper-support sense
at the cut locus. The derivative-square term from the truncation is
bounded on its annulus. Hence $\dd f_L\le CL^{-2}g$ as currents. (The
word smooth describes the scalar truncation; $f_L$ itself may not be
smooth at the cut locus.)
Let $w_L=H_{\epsilon L^2}f_L$ for a fixed small $\epsilon>0$. Gaussian
heat bounds and the Lipschitz estimate give
\[
\|w_L-f_L\|_\infty\le(C/L)\sup_x\int d_g(x,y)H_{\epsilon L^2}(x,y)
dV_g(y)\le C\sqrt{\epsilon}.
\]
Nonnegative Ricci curvature suffices for these scalar heat estimates.
The Li--Yau gradient inequality, applied to $1+H_af_L$ and $2-H_af_L$,
gives at $a=\epsilon L^2$
\[
|\Delta_gw_L|\le C_\epsilon L^{-2},\qquad
|\partial w_L|_g\le C_\epsilon L^{-1}.
\]
The first translate bounds $\partial_aH_af_L$ below, and the second
bounds it above; substituting both in the same inequality bounds the
gradient; see \cite[Theorem~1.3(i)]{LiYau}.
Lemma~\ref{lem:hodge-heat} gives the upper bound for each complex
Hessian eigenvalue; the lower trace bound then gives the lower bound
for each. Thus $|\dd w_L|_g\le C_\epsilon L^{-2}$.
Choose a smooth nondecreasing step $\theta$, zero near zero and one
near one, with constant regions wide enough to contain the above heat
error. Then $v_L=\theta(w_L)$ vanishes on $r\le L$, equals one on
$r\ge2L$, and
\[
|\partial v_L|_g\le C/L,\qquad |\dd v_L|_g\le C/L^2.
\]
The locally finite sum
\[
u=1+\sum_{j\ge0}2^jv_{2^j}
\]
is comparable to $1+r$, and at any point only a bounded number of
summands have nonzero derivatives, at scales comparable to $1+r$. This
proves \eqref{prelim:exhaustion}. Scalar cutoffs $p(u/L)$ have gradient
$O(L^{-1})$ and absolute complex Hessian $O(L^{-2})$, with support and
transition region in fixed multiples of $B(o,L)$.
\paragraph{Averaged curvature and averaged section norm.}
Put $w=\log|S|_g^2$ for a canonical section with $S(o)\ne0$. The
Poincar\'e--Lelong formula gives
\begin{equation}\label{prelim:canonical-current}
\dd w\ge\Ric(g)
\end{equation}
as currents. The truncations $w_m=\max(w,-m)$ are subharmonic. The heat
submean inequality and linear upper growth give
\[
w_m(o)\le H_{L^2}w_m(o),\qquad H_{L^2}(w_m)^+(o)\le CL.
\]
Justify the submean first on exhaustion domains and then by Gaussian
decay; the positive part grows at most linearly. For $m$ large,
$w_m(o)=w(o)$, so $H_{L^2}(w_m)^-(o)\le CL+|w(o)|$. The heat-kernel
lower bound on $B(o,L)$ is $c/\Vol B(o,L)$. Monotone convergence as
$m\to\infty$ gives the first estimate of \eqref{prelim:averages}.
Take a nonnegative cutoff equal to one on $B(o,L)$ and supported in
$B(o,CL)$, with absolute Laplacian at most $C/L^2$, as just constructed.
Integrate the trace of \eqref{prelim:canonical-current} against it and
move $\Delta_g$ onto the cutoff:
\[
\int_{B(o,L)}R_g\,dV_g\le\frac{C}{L^2}\int_{B(o,CL)}|w|\,dV_g
\le\frac CL\Vol B(o,L).
\]
The last step uses volume doubling. This proves the second estimate.
The average-scalar-curvature conclusion is
\cite[Theorem~2]{ChenZhu}; the details above record the specific
cutoffs and the absolute logarithmic estimate used later.
% =============================================================================
\section{A global flow and scalar comparison by localization}
\label{sec:flow}
We first construct the flow with estimates on fixed initial compact
sets. Completeness of its positive-time slices is a consequence of the
Harnack inequality, proved later in
Proposition~\ref{prop:harnack-completeness}.
\begin{theorem}[Localized flow]\label{thm:localized-flow}
There is a smooth K\"ahler--Ricci flow $h_t$ on $M$ for $0\le t<\infty$,
with $h_0=g$. There are smooth real functions $U,P,\rho$ such that
\begin{equation}\label{flow:identities}
\begin{aligned}
0<h_t\le g,\qquad
h_t=g-t\Ric(g)+\dd U,\qquad U(\cdot,0)=0,\\
P=U_t=\log\frac{\det h_t}{\det g},\qquad
\rho=-P\ge0,\qquad R_{h_t}=-P_t\ge0.
\end{aligned}
\end{equation}
For every finite $T$ there is $C_T$ with
\begin{equation}\label{flow:rho-average}
\sup_{0\le t\le T}\frac1{\Vol_gB_g(o,L)}
\int_{B_g(o,L)}\rho(x,t)\,dV_g(x)\le C_TL\qquad(L\ge1).
\end{equation}
At every fixed $x$, $\rho(x,t)\le C_x(1+t)$ for all $t\ge0$. On each
finite time interval there exist a smooth positive function $Q$ and a
constant $c>0$ such that
\begin{equation}\label{flow:proper-supercaloric}
Q\ge c(1+r^2),\qquad
(\partial_t-\Delta_{h_t})Q\ge c\tr_{h_t}g.
\end{equation}
Smoothness at $t=0$ means smoothness with all one-sided time derivatives.
\end{theorem}
\subsection{Complete Hermitian approximations}\label{sec:flow:approximations}
Take regular values $L\to\infty$ of the exhaustion $u$ from
Proposition~\ref{prop:initial-data} and put
$\Omega_L=\{u<L\}$. Choose a fixed smooth nonnegative $F$ on $[0,1)$,
zero on $[0,1/2]$, equal to $-2\log(1-v)$ near $v=1$, and set on
$\Omega_L$
\[
f=F(u/L),\qquad \alpha=e^fg.
\]
The estimates for $u$ imply
\begin{equation}\label{flow:conformal-bounds}
|\partial f|_\alpha\le C/L,\qquad |\dd f|_\alpha\le C/L^2.
\end{equation}
Near the boundary, $e^f=(1-u/L)^{-2}$, and the factors $(1-u/L)^{-1}$ and
$(1-u/L)^{-2}$ in the first and second derivatives are canceled by the
conformal norm. On the transition region $u\asymp L$, $|\dd u|_g\le C/u$
gives the same.
The metric $\alpha$ is complete on $\Omega_L$: approaching the regular
boundary has infinite length for the factor $(1-u/L)^{-1}$, and the
closure of $\Omega_L$ is compact. For each fixed $L$ it has bounded
geometry of every order in holomorphic charts. Take finitely many
ordinary charts on the compact closure; near a boundary point of
boundary distance comparable to $1-u/L$, shrink their coordinate radii
by a sufficiently small fixed multiple of $1-u/L$. In the rescaled
charts $\alpha$ and its inverse are uniformly bounded, and every
derivative of its coefficients is bounded, with constants allowed to
depend on $L$ and on the derivative order.
Run Chern--Ricci flow on this complete Hermitian background:
\begin{equation}\label{flow:approximation}
K=\alpha-t\Ric(\alpha)+\dd U,\qquad
U_t=P=\log\frac{\det K}{\det\alpha},\qquad U(\cdot,0)=0.
\end{equation}
Until the local limit below, $U,P$ refer to this approximation. Bounded-
geometry short-time existence is
\cite[Definition~2.1, Lemma~2.1]{LeeTam}. The estimates we produce hold
on every closed bounded-geometry interval; the resulting continuation
intervals grow with $L$.
\paragraph{Scalar evolutions.}Write $\mathscr L_K=\partial_t-\Delta_K$.
Differentiating \eqref{flow:approximation},
\begin{equation}\label{flow:scalar-evolutions}
\mathscr L_KP=-\tr_K\Ric(\alpha),\qquad
\mathscr L_KP_t=-|\Ric(K)|_K^2,
\qquad P_t(\cdot,0)\le C/L^2.
\end{equation}
The last inequality follows from
$\Ric(\alpha)=\Ric(g)-n\dd f$: initial Ricci positivity plus
\eqref{flow:conformal-bounds}. In particular
\begin{equation}\label{flow:upper-potentials}
P_t\le C/L^2,\qquad P\le Ct/L^2,\qquad U\le Ct^2/(2L^2).
\end{equation}
These are bounded maximum-principle estimates on an already existing
bounded-geometry interval.
\paragraph{A parameterized barrier.} Fix a horizon $D>0$ and suppose a
smooth $\phi\le0$ satisfies
\begin{equation}\label{flow:phi-condition}
\alpha-D\Ric(\alpha)+D\dd\phi\ge c\alpha\qquad(c>0).
\end{equation}
Define
\begin{equation}\label{flow:Q-definition}
Q=(D-t)P+U-D\phi+nt.
\end{equation}
Using $\mathscr L_KU=P-n+\tr_K(\alpha-t\Ric(\alpha))$ gives the exact
identity
\begin{equation}\label{flow:Q-comparison}
\mathscr L_KQ=\tr_K\{\alpha-D\Ric(\alpha)+D\dd\phi\}
\ge c\tr_K\alpha,\qquad Q\ge0.
\end{equation}
The initial value is $-D\phi\ge0$. For bounded $\phi$ the maximum
principle gives the last assertion directly; adding a vanishing positive
multiple of the proper $f$ is an equivalent localization. The same
argument applies to functions with downward logarithmic poles, since
$Q\to+\infty$ there.
Fix $L$ and choose finitely many canonical sections with no common zero
on $\overline{\Omega_L}$. A constant shift of
$\log\sum_i|S_i|_g^2$ is bounded, nonpositive, and has complex Hessian
at least $\Ric(g)$. It satisfies
\[
\alpha-D\Ric(\alpha)+D\dd\phi\ge\alpha+nD\dd f
\ge(1-CD/L^2)\alpha,
\]
so for fixed $D$ and large $L$ we may take $c=1/2$. Then
\eqref{flow:upper-potentials} and \eqref{flow:Q-comparison} give both
bounds for $P$ on $0\le t\le D/2$: indeed $(D-t)P\ge D\phi-U-nt$.
Integrating gives both bounds for $U$. These bounds may depend on $L$.
\subsection{The largest metric eigenvalue}\label{sec:flow:eigenvalue}
We next bound the largest eigenvalue of $K$ relative to $\alpha$. The
bound tends to one as $L\to\infty$ wherever $Q$ stays bounded
independently of $L$; the local potential estimates below will supply
that control on each fixed compact set, yielding $h_t\le g$ in the
limit.
At a point, use $g$-normal holomorphic coordinates diagonalizing $K$,
$K_{i\bar i}=\lambda_i$, and suppose $\lambda_1$ is largest. Put
$q=K_{1\bar1}$ and
\[
W=\frac{K_{1\bar1}}{e^fg_{1\bar1}}.
\]
The Rayleigh quotient in this fixed coordinate direction is a smooth
lower support for the largest relative eigenvalue, equal to it at the
point; it therefore suffices for a maximum-principle test. Wherever
$W\ge1$, direct differentiation gives
\begin{equation}\label{flow:eigenvalue-raw}
\mathscr L_K\log W
\le-\frac1q\sum_{i,j}\frac{|\partial_1K_{i\bar j}|^2}{\lambda_i\lambda_j}
+|\partial\log q|_K^2+CL^{-2}\tr_K\alpha.
\end{equation}
\emph{Derivation of \eqref{flow:eigenvalue-raw}.} Since $K-\alpha$ is
closed,
\[
\partial_1\partial_{\bar1}K_{i\bar i}-\partial_i\partial_{\bar i}K_{1\bar1}
=\partial_1\partial_{\bar1}\alpha_{i\bar i}-\partial_i\partial_{\bar i}\alpha_{1\bar1}.
\]
In the chosen coordinates the right side is $(e^f)_{1\bar1}-(e^f)_{i\bar i}$
(the second derivatives of $g$ cancel by K\"ahler symmetry). Dividing
its contraction by $q\ge e^f$ bounds the error by $CL^{-2}\tr_K\alpha$.
Differentiating the denominator $g_{1\bar1}$ contributes
$-\sum_i\lambda_i^{-1}R_g(i,\bar i,1,\bar1)\le0$. The remaining
conformal derivatives have the same bound.
For $i\ne1$, the first-derivative exchange reads
$\partial_1K_{i\bar1}=\partial_iq-e^ff_i$. The $j=1$ terms of the
negative square in \eqref{flow:eigenvalue-raw} dominate
\[
\Bigl|\partial\log q-W^{-1}\sum_{i\ne1}f_i\,dz_i\Bigr|_K^2.
\]
Expanding this square and using $\partial\log q=\partial\log W+\partial f$
at the point yields
\begin{equation}\label{flow:eigenvalue-gradient}
\mathscr L_K\log W\le CL^{-2}\tr_K\alpha
+2W^{-1}|\partial f|_K|\partial\log W|_K.
\end{equation}
\paragraph{Penalizing with $Q$.}Set $a=L^{-1}$ and
\[
J(v)=av+\log(1+av)\qquad(v\ge0).
\]
At a positive maximum of $\log W-J(Q)$,
\[
W\ge e^{aQ}(1+aQ),\qquad \partial\log W=J'(Q)\partial Q,\quad
J'\ge a,\quad J'\le2a,\quad -J''=\frac{a^2}{(1+aQ)^2}.
\]
The chain rule, \eqref{flow:Q-comparison} and
\eqref{flow:eigenvalue-gradient} give
\[
\mathscr L_K(\log W-J(Q))
\le(CL^{-2}-ca)\tr_K\alpha
+2W^{-1}J'|\partial f|_K|\partial Q|_K
+J''|\partial Q|_K^2.
\]
Young's inequality absorbs the middle term into half of
$-J''|\partial Q|_K^2$: its remaining coefficient is bounded by
\[
C\frac{W^{-2}(J')^2}{-J''}|\partial f|_K^2\le C|\partial f|_K^2\le
CL^{-2}\tr_K\alpha.
\]
For large $L$ this contradicts the maximum principle. To attain the
maximum on $\Omega_L$, first test with the additional penalty $-\delta f$;
its Hessian term is controlled by $C\delta L^{-2}\tr_K\alpha$, its
gradient terms by the same bound after Young's inequality; then let
$\delta\downarrow0$. At $t=0$, $W=1$ and $Q\ge0$. Hence
\begin{equation}\label{flow:eigenvalue-bound}
K\le e^{J(Q)}\alpha.
\end{equation}
With the bounded choice of $\phi$, $Q$ is bounded above by
\eqref{flow:upper-potentials}. Thus \eqref{flow:eigenvalue-bound} gives
an upper metric bound. The lower bound for $P$, hence for
$\det K/\det\alpha$, then bounds the least eigenvalue below.
\paragraph{Continuation in uniform holomorphic charts.}Let $\iota$ be
the real symmetric representation of a Hermitian matrix and write
\[
F(B)=\tfrac12\log\det_{\R}B,\quad
S(x,t)=\iota(\alpha-t\Ric(\alpha)),\quad
T(D^2U)=\iota(\dd U).
\]
Then $U_t-F(S+T(D^2U))=-\log\det_{\C}\alpha$. Two-sided metric
equivalence puts $S+T(D^2U)$ in a fixed compact convex positive matrix
range. The function $F$ is smooth, concave and uniformly elliptic near
this range. The map $T$ is linear, preserves nonnegativity, and has
norms comparable to the original on nonnegative real symmetric matrices.
The smooth background coefficients have uniform local H\"older bounds.
These are the hypotheses of the transformed-Hessian parabolic estimate
\cite[Theorem~5.1]{ChuRegularity}: its bound for spatial second
derivatives and first time derivative depends on $\|U\|_\infty$ and
these data, not on a prior bound for the full real Hessian. Schauder
estimates then give higher derivatives and continuation on each fixed
approximating background, as in \cite[proof of Theorem~4.1]{LeeTam}.
For initial estimates, extend $U$ by zero to negative times and use
$S(x,t_+)$ with $t_+=\max(t,0)$. Each fixed approximating solution is
smooth one-sided at zero by short-time existence. Since
$U(\cdot,0)=U_t(\cdot,0)=0$, its spatial second derivatives and first
time derivative extend continuously by zero. The extension is $C^2$ in
the parabolic sense required by Theorem~5.1 of \cite{ChuRegularity} and
solves the extended equation also at zero. The time-Lipschitz background
satisfies the H\"older hypothesis, so the same estimate crosses the
initial surface. Spatial differentiation and the linear equation for
$P$ give successive initial Schauder estimates. On each fixed compact
these estimates are uniform in $L$ once the local metric bounds below
hold and $\alpha=g$ there. Thus the approximating flows exist on
$[0,D/2]$ for all sufficiently large $L$ given the chosen $D$.
\subsection{The local limit and its potential estimates}\label{sec:flow:locallimit}
The bounds we need to pass to a limit use other choices of $\phi$. For
any canonical section $S$ from Proposition~\ref{prop:initial-data}, take
\begin{equation}\label{flow:singular-weight}
\phi=\log|S|_g^2-nf-\epsilon u^2-A_\epsilon,
\qquad 0<\epsilon\le c_0/D,\quad
A_\epsilon=C_S'(1+\epsilon^{-1}).
\end{equation}
The growth bound for $S$ makes $\phi\le0$ after choosing $C_S'$. Since
$|\dd u^2|_g\le C$, the canonical-current inequality gives
\[
\alpha-D\Ric(\alpha)+D\dd\phi
\ge\alpha-CD\epsilon g\ge\tfrac12\alpha
\]
for small $c_0$. The comparison is made away from the zeros of $S$,
where $\phi$ is smooth; the downward poles localize it away from those
zeros. Finite sums of squared norms may be used as well.
Fix a compact set on which the chosen section does not vanish. For all
large enough $L$ it lies in $\{u<L/2\}$, where $f=0$.
Equation~\eqref{flow:upper-potentials} bounds $Q$ above there by a
constant independent of $L$. Equation~\eqref{flow:Q-comparison} bounds
$P$ below there, and \eqref{flow:eigenvalue-bound} gives a local upper
metric bound with $e^{J(Q)}\to1$. The determinant lower bound then
gives a positive local lower metric bound. Finitely many nonvanishing
sections cover any compact set. Local parabolic estimates on slightly
larger compact sets give all derivatives, including the one-sided
initial estimates, uniformly in $L$. A diagonal subsequence over
compacts and over increasing horizons converges smoothly to $U,P,h$ with
$h_0=g$. The approximations have $\alpha=g$ on each fixed compact for
large $L$, so \eqref{flow:approximation} passes to the K\"ahler
equation there. The limit of \eqref{flow:eigenvalue-bound} is
$h_t\le g$; the limits of \eqref{flow:upper-potentials} give $P\le0$
and $P_t\le0$. This proves \eqref{flow:identities}.
The $Q$ comparison passes to the limit. For a weight
$\phi=\log|S|_g^2-\epsilon u^2-A_\epsilon$ and $t\le D/2$, using $U\le0$,
\begin{equation}\label{flow:rho-pointwise}
\rho(x,t)\le\frac{-D\phi(x)+nt}{D-t}
\le2\bigl(-\log|S(x)|_g^2+\epsilon u(x)^2+A_\epsilon+n\bigr).
\end{equation}
Take $D=2(T+1)$, and on a ball of radius $L$ take $\epsilon\asymp L^{-1}$
with $\epsilon\le c_0/D$. Then Proposition~\ref{prop:initial-data}
bounds the average of the right side by $C_TL$, and bounded radii are
absorbed into the constant; this proves \eqref{flow:rho-average}. At
fixed $x$, choose $S(x)\ne0$, put $D=2(t+1)$, $\epsilon=c_0/D$; the same
inequality gives $\rho(x,t)\le C_x(1+t)$.
\paragraph{A positive proper supercaloric function.}
Replace $\log|S|_g^2$ by $\Xi$. For every small enough $\epsilon>0$ the
smooth $\phi_\epsilon=\Xi-\epsilon u^2-A_\epsilon$ is nonpositive after
a finite constant shift and gives a smooth nonnegative $Q_\epsilon$
satisfying the differential inequality in \eqref{flow:Q-comparison} with
$\alpha=g$. The difference
\[
Q_\epsilon-Q_{\epsilon/2}=\frac{D\epsilon}2u^2
+D(A_\epsilon-A_{\epsilon/2})
\]
shows that $Q_\epsilon\ge cu^2-C$ because $Q_{\epsilon/2}\ge0$. Adding
a constant gives a positive $Q$ satisfying
\eqref{flow:proper-supercaloric}. This finishes the proof of
Theorem~\ref{thm:localized-flow}.
\subsection{A cutoff adapted to the evolving metric}\label{sec:flow:cutoff}
The comparison arguments below require a positive integrable weight
$\chi$ with $\dd\chi\le C\chi h_t$. A polynomial $u^{-k}$ gives only a
Hessian bound by $Cu^{-k-2}g$, and $h_t$ need not be uniformly
comparable to $g$ at infinity. We build a function $H$ whose complex
Hessian approaches $g-h_t$ at infinity. The factor
$\exp(-AH/u^2)$, for suitable $A>0$, will then contribute
$-Au^{-2}(g-h_t)$ to the logarithmic Hessian of $\chi$ and absorb errors
measured in the initial metric.
Fix $T<\infty$ and, if necessary, work on a slightly larger finite time
interval. Put
\[
\eta_t=g-h_t,\qquad a_t=\tr_g\eta_t.
\]
The forms $\eta_t$ are nonnegative, closed, bounded by $g$, with traces
$0\le a_t\le n$. The potential identity gives
\[
a_t=tR_g-\Delta_gU.
\]
Integrate against a cutoff from Section~\ref{sec:preliminaries} equal to
one on $B(o,L)$, use $-U=\int_0^t\rho(\cdot,s)\,ds$,
\eqref{prelim:averages} and \eqref{flow:rho-average}. Moving the
Laplacian onto the cutoff gives
\begin{equation}\label{flow:eta-average}
\frac1{\Vol B(o,L)}\int_{B(o,L)}a_t\,dV_g
\le C_Tt/L\qquad(L\ge1,\ 0\le t\le T).
\end{equation}
For large dyadic $L$, define
\[
H_{L,t}=-\int_0^{L^2}H_b a_t\,db.
\]
There are $\delta,\beta>0$ with, on $B(o,C_0L)$,
\begin{align}
|H_{L,t}|+L|\partial H_{L,t}|_g
&\le C_TL^{2-\delta}t^\beta,
\label{flow:heat-correction-size}\\
\dd H_{L,t}&=\eta_t+O(t/L)g,
\label{flow:heat-correction-hessian}\\
\partial_tH_{L,t}&\ge H_{L,t}/t-C_TL\qquad(t>0).
\label{flow:heat-correction-time}
\end{align}
\paragraph{Heat estimates.}
Gaussian upper bounds and relative volume comparison combined with
\eqref{flow:eta-average} give, for some fixed $p>2$,
\begin{equation}\label{flow:heat-integrand}
H_ba_t(x)\le
C_T\min\Bigl\{1,\frac tL\Bigl(\frac L{\sqrt b}\Bigr)^p\Bigr\},
\qquad x\in B(o,C_0L),\quad 0<b\le L^2.
\end{equation}
When $\sqrt b\ll L$, the ball $B(x,\sqrt b)$ may be small relative to
$B(o,L)$; relative volume comparison loses at most a fixed power of
$L/\sqrt b$, and outside $B(o,CL)$ Gaussian annuli absorb both that
power and the polynomial volume growth. The estimate $a_t\le n$ gives
the first term in the min. For the gradient, use the heat-kernel bound
\[
|\nabla_xH_b(x,y)|
\le \frac C{\sqrt b}\,
\frac{\exp(-c\,d_g(x,y)^2/b)}
{\sqrt{\Vol B(x,\sqrt b)\Vol B(y,\sqrt b)}}.
\]
It follows from \cite[Theorem~1.3, equation~(1.6)]{SoupletZhang} and
the Gaussian upper bound in \cite[Corollary~3.1]{LiYau}, absorbing
$1+d_g(x,y)^2/b$ into the Gaussian. Integrating
its absolute value against $a_t\ge0$ with the same relative-volume and
annular estimates gives $C_Tb^{-1/2}(t/L)(L/\sqrt b)^p$; increase the
fixed $p>2$ if needed. The separate bound $a_t\le n$ gives
$Cb^{-1/2}$. So the gradient obeys the minimum in
\eqref{flow:heat-integrand}, multiplied by $C/\sqrt b$. Splitting both
integrals at $b_*=L^2(t/L)^{2/p}$ shows that their bounds are
respectively
\[
C_TL^{2-2/p}t^{2/p},\qquad C_TL^{1-1/p}t^{1/p}.
\]
After increasing the finite-time constant we can take $\delta=\beta=1/p$
in \eqref{flow:heat-correction-size}. Lemma~\ref{lem:hodge-heat} gives
\[
\dd H_{L,t}=\eta_t-\mathcal H_{L^2}\eta_t.
\]
The last form is nonnegative with trace bounded by $C_Tt/L$ by
\eqref{flow:heat-integrand}; this proves
\eqref{flow:heat-correction-hessian}.
\paragraph{The time derivative.}The trace identity also gives
\begin{equation}\label{flow:H-potential-identity}
H_{L,t}=-t\int_0^{L^2}H_bR_g\,db+H_{L^2}U-U.
\end{equation}
To justify it, insert initial-metric cutoffs in
$\int_0^{L^2}H_b\Delta_gU\,db=H_{L^2}U-U$. Both $U$ and $\Delta_gU=tR_g-a_t$
have polynomially bounded absolute ball integrals, by
\eqref{flow:rho-average}, \eqref{prelim:averages}, and $0\le a_t\le n$.
Gaussian bounds for the scalar heat kernel and its gradient, together
with the absolute Laplacian bounds of the cutoffs, make the shell errors
tend to zero. The identity propagates distributionally and then smoothly
locally. Differentiating it in $t$ uses the same ball-integral bounds on
$P=-\rho$; alternatively integrate over a time interval and then
differentiate locally. Subtracting $H_{L,t}/t$ gives
\[
\partial_tH_{L,t}-H_{L,t}/t=H_{L^2}(P-U/t)-(P-U/t).
\]
Since $P_t\le0$, $P-U/t\le0$ and $0\le U/t-P\le\rho$. Its heat average
is at most $C_TL$, by \eqref{flow:rho-average} and Gaussian annular
summation. The unaveraged term has the favorable sign. This proves
\eqref{flow:heat-correction-time}.
\paragraph{Gluing.}Choose a nonnegative smooth dyadic partition of unity
$\zeta_L(u)$ with scales $L\asymp u$ and put
$H=\sum_L\zeta_L(u)H_{L,t}$ (finitely many initial scales on any fixed
compact). The partition derivatives have sizes $O(u^{-1})$ in gradient
and $O(u^{-2})$ in complex Hessian.
\eqref{flow:heat-correction-size}--\eqref{flow:heat-correction-time}
give
\begin{equation}\label{flow:glued-correction}
\begin{gathered}
|H|\le C_Tu^{2-\delta}t^\beta,\qquad
|\partial H|_g\le C_Tu^{1-\delta}t^\beta,\\
\dd H=\eta_t+O(u^{-\delta}t^\beta+t/u)g,\qquad
H_t\ge H/t-C_Tu.
\end{gathered}
\end{equation}
These are smooth in space and continuously differentiable in time,
one-sided at $t=0$, which is all the integration below uses. In fact
\eqref{flow:H-potential-identity} gives their first time derivatives
in terms of $H_{L^2}P$ and local smooth functions. The uniform absolute
ball-integral bounds for $P$ make that heat average continuous
including at $t=0$, by local convergence and uniform Gaussian tail
estimates. Spatial regularity follows from the heat equation and
\eqref{flow:heat-correction-hessian}.
\begin{lemma}[Integrable comparison cutoff]\label{flow:cutoff-lemma}
For every sufficiently large fixed integer $k$ there is a positive
$\chi(x,t)$ on $M\times[0,T]$, smooth in space and continuously
differentiable in time one-sided at the initial surface, satisfying
\begin{equation}\label{flow:cutoff-bounds}
\begin{gathered}
C_T^{-1}u^{-k}\le\chi\le C_Tu^{-k},\qquad
|\partial\chi|_g\le C_T\chi/u,\\
\dd\chi\le C_T\chi h_t,\qquad
\partial_t\chi\le C_T(1+t^{\beta-1})\chi\quad(t>0).
\end{gathered}
\end{equation}
\end{lemma}
\begin{proof}
Take
\[
\chi=u^{-k}\exp(-AH/u^2).
\]
For fixed $A$, \eqref{flow:glued-correction} gives the two-sided size
and gradient estimates. Expanding
$\chi^{-1}\dd\chi=\dd\log\chi+\partial\log\chi\otimes\bar\partial\log\chi$
gives at infinity
\[
\chi^{-1}\dd\chi\le u^{-2}\{-A(g-h_t)+C_kg+o(1)g\},
\]
uniformly on the fixed time interval. Terms containing $H/u^2$ or its
gradient are $o(u^{-2})g$ after $A$ is fixed. Choose $A>C_k+1$ and then
take $u$ large enough; the negative multiple of $g$ absorbs the
remaining error and leaves $\chi^{-1}\dd\chi\le Au^{-2}h_t$. On the
remaining compact, smoothness and positivity of $h_t$ give the stated
bound. Finally $\chi_t/\chi=-AH_t/u^2$ and
\eqref{flow:glued-correction} give
\[
\chi_t/\chi\le A|H|/(tu^2)+C_T/u\le C_T(1+t^{\beta-1}).
\]
The exponent $\beta>0$ makes this time coefficient integrable at zero.
\end{proof}
\subsection{Minimality and nonlinear scalar comparison}\label{sec:flow:scalarcmp}
\begin{proposition}[Scalar comparisons]\label{prop:scalar-comparison}
Fix finite $T>0$. For the flow of Theorem~\ref{thm:localized-flow}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\rho$ is the minimal nonnegative solution with zero initial
values of
\begin{equation}\label{flow:rho-source}
(\partial_t-\Delta_{h_t})\rho=\tr_{h_t}\Ric(g).
\end{equation}
There are nonnegative smooth supercaloric functions $\rho_j$ on
$M\times[0,T]$ with
\begin{equation}\label{flow:rho-tails}
0\le\rho_j\le\rho,\qquad \rho_j\ge\rho-C_j,\qquad
\rho_j\longrightarrow0
\end{equation}
locally uniformly on $M\times[0,T]$. Each $\rho_j$ has all spatial
and one-sided time derivatives smooth at $t=0$, bounded on compact
spacetime sets.
\item Suppose $v$ is upper semicontinuous, bounded on compact spacetime
sets, satisfies $0\le v\le\rho$, and in the upper viscosity sense
on $0<t<T$,
\begin{equation}\label{flow:nonlinear-comparison}
\partial_tv\le e^{-v}\Delta_{h_t}v+(1-e^{-v})R_{h_t}.
\end{equation}
Then $v=0$ on $M\times[0,T)$.
\end{enumerate}
Neither assertion requires completeness of $h_t$.
\end{proposition}
We first prove \textup{(i)} and the main part of \textup{(ii)}, and
state the smooth majorant as a lemma proved in the next subsection.
\begin{lemma}[Smooth majorant]\label{lem:scalar-majorant}
Let $T>0$ and let $v$ be upper semicontinuous, bounded on compact
spacetime sets, and satisfy $0\le v\le\rho$ on $M\times[0,T]$. Suppose
$v$ satisfies \eqref{flow:nonlinear-comparison} in the upper viscosity
sense on $0<t<T$. There is $w$ smooth on $M\times(0,T)$ and continuous
on $M\times[0,T)$, such that
\[
v\le w\le\rho,\qquad
w_t=e^{-w}\Delta_{h_t}w+(1-e^{-w})R_{h_t},\qquad
w(\cdot,0)=0.
\]
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:scalar-comparison}]
The Ricci identity $\Ric(h_t)=\Ric(g)+\dd\rho$, traced with $h_t$,
gives \eqref{flow:rho-source} because $\rho_t=R_{h_t}$ and the source is
nonnegative. Solve the source equation with zero initial and lateral
Dirichlet data on a smooth relatively compact exhaustion: first a weak
energy solution \cite[Chapter~III, Theorem~4.1]{LSU}, then local
parabolic regularity. A source need not vanish at the initial lateral
corner, so full corner compatibility is not asserted. The construction
and its energy estimates pass between a finite set of coordinate charts
and require no completeness of the evolving metric. Domain comparison
makes these solutions increase, and comparison with $\rho$ bounds them
above. Their limit $\rho_{\min}$ is the minimal nonnegative potential.
Local estimates give a smooth solution for $t>0$ and continuous zero
initial values, since it is bounded by $\rho=O_K(t)$ on each compact.
The difference $z=\rho-\rho_{\min}$ satisfies $0\le z\le\rho$,
$z_t=\Delta_{h_t}z$, $z(\cdot,0)=0$.
\emph{Mass argument.} Use $\chi$ from Lemma~\ref{flow:cutoff-lemma} with
$k>2n+3$ and a cutoff $p_N=p(u/N)$, equal to one on $u\le N$ and zero on
$u\ge2N$. Set $I_N(t)=\int_Mz\chi p_N\,dV_{h_t}$. Since
$\partial_tdV_{h_t}=-R_{h_t}dV_{h_t}$ and $R_{h_t}\ge0$, integration by
parts on the compact support of $p_N$ gives
\begin{equation}\label{flow:linear-mass}
I_N'(t)\le C_T(1+t^{\beta-1})I_N(t)+E_N(t),\qquad
|E_N(t)|\le C_TN^{2n-1-k}.
\end{equation}
The error carries the derivatives falling on $p_N$; its complex
Hessian, including gradient cross terms with $\chi$, has $g$-norm at
most $C_TN^{-k-2}$ on $N\le u\le2N$. The pointwise cofactor identity and
$0<h_t\le g$ give, for any real $(1,1)$-form $B$,
\begin{equation}\label{flow:cofactor}
|\tr_{h_t}B|\,dV_{h_t}\le C_n|B|_g\,dV_g.
\end{equation}
Indeed diagonalize $h_t$ against $g$; the coefficient multiplying each
diagonal entry of $B$ is the product of the other $n-1$ eigenvalues,
each at most one. This covers $n=1$ too.
Equation~\eqref{flow:rho-average} bounds the integral of $z$ on the
shell by $C_TN^{2n+1}$, proving the error bound.
The mass $I=\int z\chi\,dV_h$ is finite. Its tails tend to zero
uniformly in $t$ by $z\le\rho$, $dV_h\le dV_g$, and the dyadic shell
estimate. Its initial limit is zero by local convergence and this tail
control. Integrate \eqref{flow:linear-mass} from a positive initial
time, send $N\to\infty$, then send that initial time to zero, to get
$I(t)\le\int_0^tC_T(1+s^{\beta-1})I(s)\,ds$. Gronwall gives $I=0$;
positivity of $\chi$ gives $z=0$, hence $\rho=\rho_{\min}$.
\emph{Tails.} Choose smooth spatial cutoffs $0\le\theta_j\le1$
increasing to one, each compactly supported and eventually equal to one
near each fixed compact. Let $\rho_j$ be the minimal zero-data
potential of $F_j=(1-\theta_j)\tr_{h_t}\Ric(g)$. It is nonnegative and
supercaloric. Linearity on each exhaustion domain and monotone limits
decompose $\rho$ into this exterior-source potential and the minimal
potential of $\theta_j\tr_h\Ric(g)$; the latter is at most
$T\sup_{M\times[0,T]}\theta_j\tr_h\Ric(g)<\infty$ by the scalar maximum
principle on the domains. This gives $\rho_j\ge\rho-C_j$. Monotone
convergence of the nonnegative Dirichlet Green integrals, first in the
domains and then in the source cutoffs, gives $\rho_j\downarrow0$
pointwise.
\emph{Initial regularity.} On each fixed compact neighborhood the
coefficients of $\Delta_{h_t}$ and the source $F_j$ are smooth and
uniformly parabolic through $t=0$. The domination
$0\le\rho_j\le\rho=O_K(t)$ supplies continuous zero initial values.
Extend coefficients and source from a coordinate neighborhood to a
smooth uniformly parabolic Cauchy problem and solve with zero initial
\cite[Chapter~IV, Theorem~5.1]{LSU}. Subtract the local particular
solution $\rho_j$ from a smaller neighborhood; the difference is $O(t)$
there and solves the homogeneous equation for $t>0$. A spatially
localized energy estimate from time $\delta>0$, then $\delta\downarrow0$,
puts the difference in the local weak energy class through zero, and
the squared $L^2$ norm of its initial value on the fixed compact is
$O(\delta^2)$. Extend the difference by zero to negative times and
extend the coefficients smoothly; its zero initial trace removes a
distributional boundary term, so it solves the homogeneous equation
across the initial surface, and interior regularity
\cite[Chapter~III, Theorem~12.1]{LSU} makes it smooth there. Adding
back the particular solution gives the one-sided smooth jets.
Then $\partial_t\rho_j(x,0)=F_j(x,0)$ and higher initial derivatives
are determined recursively from
$(\rho_j)_t=\Delta_{h_t}\rho_j+F_j$, smooth in $x$. Dini's theorem on
compact spacetime sets gives the local uniform convergence in
\eqref{flow:rho-tails}. An auxiliary smooth extension across $t=0$ can
be chosen to match these jets if needed; extension by zero to negative
time is not asserted.
\emph{The nonlinear comparison.} Let $w$ be the minorant from
Lemma~\ref{lem:scalar-majorant}: $v\le w\le\rho$, and $w$ solves
\eqref{flow:nonlinear-comparison} with equality. Multiplying by $e^w$
gives the volume-form identity
\begin{equation}\label{flow:nonlinear-volume}
\partial_t[(e^w-1)dV_{h_t}]=(\Delta_{h_t}w)dV_{h_t},
\end{equation}
where the cancellation uses $\partial_tdV_h=-R_hdV_h$ and
$e^w(1-e^{-w})R_h=(e^w-1)R_h$.
Integrate \eqref{flow:nonlinear-volume} against $\chi p_N$ and move the
Laplacian onto that compactly supported function. Since $w\le e^w-1$ and
$\Delta_h\chi\le C_T\chi$, the interior term is bounded by a constant
times $M_N(t)=\int_M(e^w-1)\chi p_N\,dV_h$; the time derivative of
$\chi$ is bounded by $C_T(1+t^{\beta-1})\chi$; the exterior error is
$O(N^{2n-1-k})$ by $w\le\rho$ and \eqref{flow:cofactor}. Moreover
\[
0\le(e^w-1)dV_h\le e^\rho dV_h=dV_g.
\]
The weighted mass is finite with uniformly small tails and zero initial
limit (by dominated convergence on compacts and $\chi\asymp u^{-k}$).
The same limiting Gronwall argument forces it to vanish, so $w=0$; then
$0\le v\le w$ gives \textup{(ii)}.
\end{proof}
\subsection{Constructing the smooth majorant}\label{sec:flow:majorant}
\begin{proof}[Proof of Lemma~\ref{lem:scalar-majorant}]
We build the majorant on relatively compact domains and pass to a limit.
\emph{Barriers.} On a smooth relatively compact $\Omega$ and for time
$<T$, consider equality in \eqref{flow:nonlinear-comparison}. Zero and
$\rho$ are respectively sub- and supersolution; check the latter using
$R_{h_t}-\Delta_{h_t}\rho=\tr_{h_t}\Ric(g)\ge0$. The function
$\rho+2\alpha$ is also a supersolution for every $\alpha>0$.
Take constant initial data $\alpha$. On $\partial\Omega$ choose a smooth
$\zeta:[0,\infty)\to[0,1]$ equal to one near zero and supported in
$[0,1]$, and put
\[
\varphi(x,t)=\rho(x,t)+\alpha-e^{-\alpha}R_g(x)t\zeta(t/\delta).
\]
For $\delta\sup_{\partial\Omega}R_g\le\alpha/2$, this lies between
$\rho+\alpha/2$ and $\rho+\alpha$. It has $\varphi(\cdot,0)=\alpha$ and
$\varphi_t(\cdot,0)=(1-e^{-\alpha})R_g$, so the value and first
equation compatibility conditions are exact.
\emph{Local existence on the fixed domain.}
Use the compatible linear Dirichlet Schauder theorem
\cite[Chapter~IV, Theorem~5.2]{LSU} and its uniform small-time inverse
\cite[Chapter~IV, Theorem~5.4]{LSU}. These Euclidean-domain statements
give the compact-manifold form by the following chart-summation
argument. Choose a partition of unity $\chi_i$ and cutoffs $\psi_i=1$
near $\supp\chi_i$ in finitely many interior and boundary charts.
Local domains are smoothly capped outside these supports, and $\psi_i$
vanishes near every artificial boundary. If $E_i$ is the local zero-data
inverse, $Sf=\sum_i\psi_iE_i(\chi_if)$ satisfies
$LSf=f+\mathcal Rf$ for the global linear operator $L$. The remainder
consists of first- and zeroth-order commutators
$[L,\psi_i]E_i(\chi_if)$. On the source space $f(\cdot,0)=0$, the
local solutions have zero initial trace. Parabolic interpolation and
the uniform inverse bounds give
$\|\mathcal R\|_{C^{\beta,\beta/2}\to C^{\beta,\beta/2}}\le C\epsilon^\theta$
on a time interval of length $\epsilon$, with $\theta>0$; for
$0<\beta<1$, $\theta=(1-\beta)/2$ works. The remainder preserves the
source space, so $S(I+\mathcal R)^{-1}$ is the required global inverse
on a small interval. All constants may depend on $\Omega$.
\emph{Compatible boundary data.}
Extend the boundary data to a smooth $p$ with $p(x,0)=\alpha$ and
$p_t(x,0)=c(x):=(1-e^{-\alpha})R_g(x)$ throughout $\Omega$. In a smooth
boundary collar with projection $\pi$ and cutoff $\vartheta=1$ near the
boundary, take
\[
p(x,t)=\alpha+tc(x)+\vartheta(x)\{\varphi(\pi(x),t)-\alpha-tc(\pi(x))\},
\]
and $\alpha+tc(x)$ outside the collar. The bracket and its first time
derivative vanish at zero.
Put $L=\partial_t-e^{-p}\Delta_h$ and seek
$v_{\Omega,\alpha}=p+z$ with zero initial and lateral data for $z$:
\[
Lz=\mathcal N(z):=(e^{-p-z}-e^{-p})\Delta_h(p+z)
+(1-e^{-p-z})R_h-Lp.
\]
For every such $z\in C^{2+\beta,1+\beta/2}$,
$\mathcal N(z)(x,0)=0$. The linear inverse applies to compatible
sources. If $z(\cdot,0)=0$, then $\|z\|_\infty\le\epsilon\|z_t\|_\infty$;
spatial interpolation and the time H\"older quotient give
\[
\|z\|_{C^{\beta,\beta/2}}
\le C\epsilon^{1-\beta/2}\|z\|_{C^{2+\beta,1+\beta/2}}.
\]
On a fixed ball in the latter space, product and composition estimates
therefore give
\[
\|\mathcal N(z)-\mathcal N(w)\|_{C^{\beta,\beta/2}}
\le C\epsilon^{1-\beta/2}\|z-w\|_{C^{2+\beta,1+\beta/2}}.
\]
The coefficient of every highest-derivative difference has this small
factor. $\mathcal N(0)$ is smooth and vanishes at $t=0$, so its
$C^{\beta,\beta/2}$ norm tends to zero with $\epsilon$. The linear
inverse and the contraction theorem give a local Dirichlet solution;
classical comparison with the barriers gives
$0\le v_{\Omega,\alpha}\le\rho+2\alpha$.
\emph{Continuation.} Write $\Delta_h=a^{ij}\partial_i\partial_j+b^j\partial_j$
in fixed real coordinates and put $q=e^{v_{\Omega,\alpha}}$. Directly,
\[
q_t-\partial_i\Bigl(\frac{a^{ij}}q\partial_jq\Bigr)
=\frac{b^j-\partial_ia^{ij}}q\partial_jq+(q-1)R_h.
\]
The barriers bound $q$ above and away from zero. The leading matrix is
uniformly elliptic, and the drift and forcing are bounded independently
of $\nabla q$ on the fixed cylinder. Boundary and interior H\"older
estimates apply \cite[Chapter~III, Theorem~10.1]{LSU}; the
exterior-measure condition holds for the smooth compact boundary. Hence
$v_{\Omega,\alpha}$ is H\"older up to the parabolic boundary. Its
original equation now has H\"older coefficients and source, so
\cite[Chapter~IV, Theorem~5.2]{LSU} gives a controlled
$C^{2+\gamma,1+\gamma/2}$ bound on the closed slab, $0<\gamma<1$. In
particular its second spatial derivatives are bounded.
On slabs separated from the initial surface, differentiated boundary
Schauder estimates give all higher bounds. At a hypothetical positive
maximal time the solution has smooth limiting initial data satisfying
the boundary equation jets, and the same local construction restarts
it. This proves existence up to the prescribed finite horizon. No
constants near the remote boundaries need be uniform as $\Omega$
exhausts $M$.
\emph{Viscosity comparison.} This gives $v\le v_{\Omega,\alpha}$. At a
positive interior maximum of $e^{-Ct}(v-v_{\Omega,\alpha})$, the test
has the same spatial Hessian as $v_{\Omega,\alpha}$; the difference of
the right sides is bounded by a constant times
$v-v_{\Omega,\alpha}$ because the derivative in the value variable is
$e^{-v}(R_h-\Delta_hv_{\Omega,\alpha})$, bounded on the compact slab.
Choose $C$ larger than that bound and use a strict increasing time
penalty. The initial and lateral gaps are positive, so no boundary
maximum interferes. This is the usual viscosity argument
\cite{CIL}.
\emph{Limit on $M$.} Exhaust $M$, let $\alpha\downarrow0$, and use local
uniformly parabolic estimates to extract a smooth interior limit $w$:
\[
v\le w\le\rho,\qquad
w_t=e^{-w}\Delta_hw+(1-e^{-w})R_h,\qquad w(\cdot,0)=0.
\]
The limit at the initial surface follows from the barriers on each
compact; no uniform estimates near the remote lateral boundaries are
required.
\end{proof}
% ---------------------------------------------------------------------------
% End of Section 3. Next: the cotangent envelope (discs, variation,
% ellipsoids), then charts, dynamics, uniformization, bibliography.
% =============================================================================
\section{The cotangent envelope: compactness and frames}
\label{sec:discs}
We study the flow through boundary averages of its dual squared norm on
holomorphic cotangent discs. Let $\Delta=\{z\in\C:|z|<1\}$ and $dA$
Euclidean area on $\Delta$, and write $\pi:Y=T^{*(1,0)}M\to M$. Put
\[
N(x,\xi,t)=|\xi|^2_{h_t^{-1}}.
\]
Then $N(\cdot,t)\ge N(\cdot,0)$ by Theorem~\ref{thm:localized-flow}. The
function $N(\cdot,0)$ is plurisubharmonic: the squared norm on the total
space of a Griffiths-nonpositive bundle has this property, and the dual
of $T^{1,0}M$ carries that curvature sign.
Fix $T<\infty$. Let physical time at a disc center be $s$. For $y\in Y$
and $0<s<T$, let $v^{\mathrm{raw}}(y,s)$ be the infimum of
\begin{equation}\label{var:envelope}
\bigl\langle N(f(\zeta), s e^{w(\zeta)+\overline{w(\zeta)}})\bigr\rangle,
\qquad (f(0),w(0))=(y,0),
\end{equation}
over holomorphic discs $f:\overline\Delta\to Y$ and $w:\overline\Delta\to\C$,
smooth on the closed unit disc, with $s e^{w+\bar w}<T$ there. Brackets
mean normalized circle average. Let $v$ be the lower semicontinuous
regularization in $(y,s)$. Constant discs, plurisubharmonicity of
$N(\cdot,0)$, and $N(\cdot,t)\ge N(\cdot,0)$ give
\begin{equation}\label{var:sandwich}
N(y,0)\le v(y,s)\le v^{\mathrm{raw}}(y,s)\le N(y,s).
\end{equation}
Multiplying the fiber component of any competing disc by a nonzero
complex number shows $v$ is homogeneous of degree two in the fiber. The
assertion at the zero section follows from \eqref{var:sandwich}.
Minimizing boundary averages over discs with a prescribed center is the
Poisson-envelope construction of Poletsky and Rosay
\cite{Poletsky1991,Rosay2003}; see also
\cite[Sections~1--2]{LarussonSigurdsson} for the manifold framework. Here
we prove the differential inequality needed for the evolving norm
directly, with the compactness and quantitative deformation estimates
stated below.
\emph{Goal and obstacle.} We prove $v=N$. This gives the submean
inequality for $N(x,\xi,e^{w+\bar w})$, hence its joint
plurisubharmonicity. The competitors in \eqref{var:envelope} have no
common boundary compact and no common area bound. In
Section~\ref{sec:variation} we localize almost-minimizing discs by
boundary walls and a positive area penalty; the compact containing
their full images may then depend on the penalty. With centers in a
fixed compact coordinate region, the frame estimates used in the
differential inequality must depend on boundary and center data,
independently of the area bound and the compact containing the
interiors. That is what this Section establishes, together with the
cotangent evolution identity used in the estimate. All compactness
arguments use the complete initial metric $g$.
\subsection{The symplectic Hessian operator}\label{sec:discs:symplectic}
The canonical holomorphic bivector on $Y$ is
\[
\mathcal J=\sum_{i=1}^n\frac{\partial}{\partial x_i}\wedge
\frac{\partial}{\partial\xi_i}.
\]
We use column matrices for Hermitian forms, so a form with matrix $W$
has quadratic value $v^*Wv$. In a frame $e_1,\ldots,e_{2n}$, write
$\mathcal J=\sum_{A<B}J^{AB}e_A\wedge e_B$ and define
\begin{equation}\label{disc:p-definition}
p(W,\mathcal J)=
\sum_{A<B,\,C<D}\overline{J^{AB}}J^{CD}(W_{AC}W_{BD}-W_{AD}W_{BC}).
\end{equation}
This is the induced exterior-square squared norm when $W$ is positive,
and a real polynomial for every Hermitian $W$. It is independent of
frame. In canonical cotangent coordinates abbreviate it to $p(W)$. If
$W=\left(\begin{smallmatrix}A&B\\B^*&C\end{smallmatrix}\right)$, then
\begin{equation}\label{disc:p-block}
p(W)=\tr(AC^{\mathsf T})-\sum_{i,j}B_{ij}\overline{B_{ji}}.
\end{equation}
The transpose records the tangent--cotangent pairing. In particular, if
tangent coefficients have metric matrix $h$, the column matrix of the
dual norm is $b=h^{-\mathsf T}$.
\begin{lemma}[Symplectic evolution]\label{lem:symplectic-evolution}
For the flow of Theorem~\ref{thm:localized-flow},
\begin{equation}\label{disc:symplectic-evolution}
\partial_tN=p(\dd_YN).
\end{equation}
For every real smooth base function $d=d(x,t)$, with $t$ held fixed in
the spatial Hessian,
\begin{equation}\label{disc:base-addition}
p(\dd_Y(N+d))=\partial_tN+\Delta_{h_t}d.
\end{equation}
These identities do not assume that $h_t$ is complete or has
nonnegative curvature at positive time.
\end{lemma}
\begin{proof}
Fix $(x,t)$ and use K\"ahler-normal coordinates for $h_t$ there. Then
$h_t=b_t=I$, all first spatial derivatives of $b_t$ vanish, and the
Hessian of $N$ has blocks
\[
\dd_YN=\begin{pmatrix}\mathcal R_\xi&0\\0&I\end{pmatrix}.
\]
The horizontal form $\mathcal R_\xi$ is the bisectional curvature form of
$h_t$ with one pair of indices contracted against the metric dual of
$\xi$. Differentiating $b=h^{-\mathsf T}$ twice at a normal point gives
negative second derivatives of $h$, transposed, which are exactly those
curvature components. The K\"ahler symmetries give
\[
\tr\mathcal R_\xi=\xi^*\Ric(h_t)^{\mathsf T}\xi.
\]
On the other hand $\partial_th=-\Ric(h)$ implies
$\partial_tb=h^{-\mathsf T}\Ric(h)^{\mathsf T}h^{-\mathsf T}$.
Equation~\eqref{disc:p-block} now proves \eqref{disc:symplectic-evolution}.
Adding a base Hessian changes only the horizontal block, and its
contribution to $p$ is $\tr((\dd_xd)b^{\mathsf T})=\tr(h_t^{-1}\dd_xd)$,
proving \eqref{disc:base-addition}. The calculation is tensorial, so the
normal-coordinate verification proves both identities everywhere.
\end{proof}
\subsection{Compact boundary and bounded area}\label{sec:discs:compact}
\begin{proposition}[Compactness of discs]\label{prop:disc-compactness}
For each compact $K\subset M$ and each $A_0<\infty$ there is a compact
$K'\subset M$ such that every holomorphic disc
$f:\overline\Delta\to M$, smooth up to the circle and satisfying
$f(\partial\Delta)\subset K$ and $\int_\Delta f^*\omega_g\le A_0$,
has $f(\overline\Delta)\subset K'$. Every sequence of such discs has a
subsequence converging holomorphically on compact subsets of $\Delta$.
The same statements hold for discs in $Y$ with compact boundary in $Y$
and with $\int_\Delta(\pi\circ f)^*\omega_g\le A_0$. The resulting
compact may depend on $K$ and $A_0$.
\end{proposition}
The area bound first places part of each disc in a fixed compact. An
inverse-Jacobian estimate for the global injective immersion then
propagates this control away from isolated possible singularities of a
limit; properness of the immersion is not assumed. The
extra-integrability lemma removes those punctures, and a subharmonic
compactness lemma carries the initial control.
\begin{lemma}[Extra integrability]\label{lem:extra-integrability}
Let $(E,k)$ be a smooth Hermitian holomorphic vector bundle on
$\Delta^*=\{0<|z|<1\}$ with Griffiths nonnegative curvature. Suppose
$A:E\to\Delta^*\times\C^q$ is holomorphic, pointwise injective, and has
bounded operator norm. If $a$ is a holomorphic section with
$\int_{\Delta^*}|a|_k^2\,dA<\infty$, then there are $r_0,\sigma>0$ such
that
\begin{equation}\label{disc:extra-integrability}
\int_{0<|z|<r_0}|a|_k^2|z|^{-\sigma}\,dA<\infty.
\end{equation}
The exponent may depend on the section and the bundle data.
\end{lemma}
\begin{proof}
Choose a fixed smooth $\chi:\R\to[0,1]$ zero on $(-\infty,-2]$ and one
on $[-1,\infty)$, and set
\[
\chi_m(z)=\chi(m^{-1}\log|z|),\qquad
\phi_m(z)=\frac2m\log|z|+|z|^{1/m}.
\]
Writing $r=|z|$, one has
\[
|\dbar\chi_m|\le\frac{C}{mr},\qquad
(\phi_m)_{z\bar z}=\frac1{4m^2}r^{1/m-2}.
\]
On a complex curve Griffiths and Nakano nonnegativity coincide. Apply the
bundle-valued H\"ormander estimate to $E$-valued $(1,0)$-forms with the
metric $ke^{-\phi_m}$ and source $a\,\dbar\chi_m\wedge dz$; see
\cite{Hormander} and \cite[Theorem~5.1, Remark~4.5]{Demailly}. One may
use any complete auxiliary K\"ahler metric on $\Delta^*$; in dimension
one its conformal factor cancels both from the norm of the unknown top
form times volume and from the curvature-inverse source pairing times
volume. Dropping the nonnegative curvature of $k$ gives
$v_m$ with $\dbar v_m=a\,\dbar\chi_m$ and
\begin{align}
\int_{\Delta^*}|v_m|_k^2e^{-\phi_m}\,dA
&\le C\int_{e^{-2m}<r<e^{-m}}
|a|_k^2r^{-3/m}e^{-r^{1/m}}\,dA,\label{disc:weighted-solve}\\
\int_{\Delta^*}|v_m|_k^2r^{-2/m}\,dA
&\le C\int_{e^{-2m}<r<e^{-m}}|a|_k^2\,dA\longrightarrow0.
\label{disc:correction-small}
\end{align}
For the second estimate use $r^{-3/m}\le e^6$ on the source annulus and
$e^{-1}\le e^{-r^{1/m}}\le1$; constants are independent of $m$. The
estimate concerns smooth metrics on the punctured domain; no curvature
extension at zero is asserted.
The sections $a_m=\chi_ma-v_m$ are holomorphic and converge to $a$ in
unweighted $L^2$. Near zero they equal $-v_m$, so each has a positive
extra-integrability exponent, namely $2/m$. Since $A$ is bounded,
$Aa_m$ and $Aa$ are ordinary $L^2$ holomorphic vector functions on the
punctured disc, and their negative Laurent coefficients vanish by $L^2$
integrability; hence they extend across zero. Their $L^2$ convergence
and interior Cauchy estimates give convergence of every finite jet at
zero.
\emph{Closing the exponent.} Let $R=\mathcal O_{\C,0}$, with maximal
ideal $\mathfrak m=(z)$, and let $S\subset R^q$ be the set of germs $Ab$
where $b$ is holomorphic on some punctured neighborhood and
$\int|b|_k^2r^{-\varepsilon}\,dA<\infty$ for some $\varepsilon>0$. These
images extend across zero by the same $L^2$ argument. The set $S$ is an
$R$-submodule: a sum uses the smaller of the two exponents, and
multiplication by a holomorphic germ is bounded after shrinking the
neighborhood. The ring $R$ is Noetherian, so $S$ is a submodule of a
finite free module and $Q=R^q/S$ is a finite module. For each $k$ the
image of $S$ in $R^q/\mathfrak m^kR^q$ is a finite-dimensional closed
subspace. Jet convergence of $Aa_m\in S$ therefore gives
$Aa\in S+\mathfrak m^kR^q$ for every $k$. Krull's intersection theorem
for the finite local module $Q$ gives
$\bigcap_k\mathfrak m^kQ=0$ \cite[Lemma~10.51.4]{StacksKrull}. Thus
$Aa\in S$, so it is the image of one section $b$ with some positive
extra exponent. Pointwise injectivity of $A$ implies $a=b$ on a
sufficiently small punctured neighborhood, proving
\eqref{disc:extra-integrability}. No common exponent for the $a_m$ was
used.
\end{proof}
Let $F=(F_1,\ldots,F_N):M\to\C^N$ be the global injective holomorphic
immersion of Proposition~\ref{prop:initial-data}. Set
\[
\psi=\log\sum_{|I|=n}|dF_I|_g^2,
\qquad dF_I=dF_{i_1}\wedge\cdots\wedge dF_{i_n}.
\]
Immersivity makes $\psi$ smooth. Along any holomorphic disc, $\psi$ is
subharmonic: each $dF_I$ is a holomorphic canonical form, whose
logarithmic squared norm is plurisubharmonic under the curvature
hypothesis, and the log-sum inherits the property. Similarly
$|dF|_g^2=\sum_i|dF_i|_g^2$ is plurisubharmonic.
\begin{lemma}[Compactness with a lower anchor]\label{lem:psh-compactness-bridge}
Let $\Omega_j$ be increasing open subsets exhausting a connected complex
manifold $X$, and let $u_j$ be plurisubharmonic on $\Omega_j$. Suppose
that for each compact $K\subset X$, the $u_j$ are uniformly bounded
above on $K$ once $K\subset\Omega_j$. Every sequence has a subsequence
where either $u_j\to-\infty$ locally uniformly from above, or $u_j$
converges in $L^1_{\mathrm{loc}}$ to a plurisubharmonic function. A
lower bound $u_j(z_j)\ge-A$ at points $z_j$ in a fixed compact set
excludes the first alternative along that sequence. In particular,
lower bounds on subsets of a fixed compact set of uniformly positive
measure suffice. If the $L^1_{\mathrm{loc}}$ limit is the constant $c$,
then
\[
\limsup_j\sup_Ku_j\le c
\]
for every compact $K\subset X$.
\end{lemma}
\begin{proof}
\emph{Lower anchor.} Pass to a subsequence with $z_j\to z_*$ and work in
a coordinate ball about $z_*$. On a slightly larger ball choose a
common upper bound $C$ and set $w_j=C-u_j$; these are nonnegative
superharmonic. The supermean inequality at $z_j$ bounds their integrals
on a ball of fixed radius about $z_j$ by its volume times $C+A$. A
fixed smaller ball about $z_*$ is eventually contained in those balls,
so the $L^1$ norms there are bounded.
\emph{Propagation.} Such a bound propagates along finite chains of
overlapping coordinate balls. In a fixed overlap of positive measure,
the integral bound supplies a point at which $w_j$ is bounded by its
average over the overlap. Applying the supermean inequality on a larger
ball centered at that point, still contained in the coordinate
neighborhood, bounds the integral on the next smaller ball. On each
fixed compact overlap the Euclidean volume measures in the two
coordinate charts are uniformly comparable; increasing the local upper
bound $C$ when moving between charts only adds a fixed constant.
Connectedness and a finite covering give $L^1$ bounds on every fixed
compact subset of $X$.
\emph{Compactness.} Families of subharmonic functions bounded in
$L^1_{\mathrm{loc}}$ are relatively compact there
\cite[Chapter~I, Proposition~4.21]{DemaillyBook}. Apply in coordinate
balls and take a diagonal subsequence. Positivity of the complex
Hessians passes to the distributional limit, so its upper
semicontinuous representative is plurisubharmonic. If no subsequence
has a lower anchor in any fixed compact set, then the suprema on each
member of a compact exhaustion tend to $-\infty$; that is the other
alternative. The assertion for exhausting domains follows because each
finite chain of balls lies in every sufficiently late $\Omega_j$.
\emph{Constant limit.} Suppose the limit is the constant $c$. In a
coordinate ball whose concentric enlargement is relatively compact,
apply the submean inequality on a fixed-radius ball centered at each
point of the smaller ball: it bounds $u_j-c$ from above by a fixed
constant times the $L^1$ norm of $u_j-c$ on the enlargement. This norm
tends to zero. A finite covering proves the asserted upper bound on $K$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:disc-compactness}]
\emph{Step 1: anchors from the area bound.}
The maximum principle gives uniform bounds
\begin{equation}\label{disc:boundary-upper}
|F\circ f_l|\le C_K,\qquad |dF|_g\circ f_l\le C_K,\qquad
\psi_l:=\psi\circ f_l\le C_K.
\end{equation}
The area bound controls the energy up to a fixed normalization. In
particular, the functions
\[
E_l(\theta)=\int_{1/2}^1|\partial_rf_l(re^{i\theta})|_g^2\,r\,dr
\]
have uniformly bounded angular integrals. On a set of angles of measure
bounded below, $E_l(\theta)\le C(A_0+1)$. For such $\theta$ and every
$r\in[1/2,3/4]$, Cauchy--Schwarz gives
\[
\dist_g(f_l(re^{i\theta}),K)
\le E_l(\theta)^{1/2}\Bigl(\int_{1/2}^1\frac{dr}r\Bigr)^{1/2}.
\]
Completeness and Hopf--Rinow put these images in a fixed compact. Thus
on a subset of the fixed annulus of area bounded below, $\psi_l$ has a
fixed lower bound.
Lemma~\ref{lem:psh-compactness-bridge}, together with
\eqref{disc:boundary-upper}, now gives a subsequence converging in
$L^1_{\mathrm{loc}}(\Delta)$ to a subharmonic $\psi_\infty$; the lower
bounds on the anchor subsets exclude collapse to $-\infty$. We retain
their positive-area property for the later step that avoids neighborhoods
of the possible atoms. Let
$\mu_l=(2\pi)^{-1}\Delta_{\mathrm{eucl}}\psi_l$ and
$\mu=(2\pi)^{-1}\Delta_{\mathrm{eucl}}\psi_\infty$; then
$\mu_l\to\mu$ weakly on compact subsets.
\emph{Step 2: inverse-Jacobian estimate.}
Let $Z=\{z:\mu(\{z\})\ge1\}$, a locally finite subset of $\Delta$. Near
any point outside $Z$, choose a small disc $D$ whose boundary has zero
$\mu$-measure and for which $\mu(D)<b<4/3$. For large $l$,
$\mu_l(D)<b$. The Riesz decomposition on $D$
\cite[Chapter~I, Proposition~4.22]{DemaillyBook} is
\[
\psi_l=h_l+\int_D\log|z-\zeta|\,d\mu_l(\zeta).
\]
On each smaller concentric disc the harmonic terms $h_l$ are bounded in
absolute value: their $L^1$ norms are bounded by $L^1$ convergence of
$\psi_l$, the bounded masses, and local integrability of the
logarithmic kernel; harmonic interior estimates then apply. For a
positive measure $\nu$ of mass $b_l\le b$, Jensen's inequality gives,
with the zero-mass case read separately,
\[
\exp\Bigl(-p\int_D\log|z-\zeta|\,d\nu(\zeta)\Bigr)
\le\frac1{b_l}\int_D|z-\zeta|^{-pb_l}\,d\nu(\zeta),
\qquad p=\tfrac32.
\]
Integration in $z$ gives a uniform bound, since $pb<2$. Therefore
$e^{-\psi_l}$ is locally bounded in $L^{3/2}(\Delta\setminus Z)$.
Let $s_1\le\cdots\le s_n$ be the singular values of $dF$ measured from
$g$ to the Euclidean metric. Cauchy--Binet and
\eqref{disc:boundary-upper} give
\[
e^{\psi_l}=\prod_i s_i(f_l)^2,\qquad
s_1(f_l)^{-2}\le C_Ke^{-\psi_l}.
\]
Consequently
\begin{equation}\label{disc:inverse-jacobian}
|f_l'|_g^2\le C_Ke^{-\psi_l}|(F\circ f_l)'|^2.
\end{equation}
The bounded holomorphic $F\circ f_l$ have uniformly bounded derivatives
on smaller discs, so $df_l$ is locally bounded in $L^3$ off $Z$. For
$p\in M$ the function $v_{l,p}(z)=\dist_g(f_l(z),p)$ has weak gradient
bounded by $|df_l|_g$. Morrey's local gradient-only estimate in real
dimension two, with exponent $1-2/3=1/3$
\cite[Section~5.6.2, Theorem~4 and the Remark on p.~283]{EvansPDE},
applies; its constant depends on the nested domain discs, not on $p$,
and no bound on the values of $v_{l,p}$ is needed. Taking $p=f_l(w)$
gives the uniform local metric H\"older estimate.
\emph{Step 3: propagating from the anchors.}
Remove small neighborhoods of the finitely many points of $Z$ in the
anchor annulus, with total area smaller than half the anchor lower
bound. Choose anchor points in the remaining compact set and pass to a
convergent subsequence of their domain positions. The local H\"older
estimate puts neighboring images in a fixed bounded $g$-neighborhood of
the anchor compact. Any compact subset of the connected
$\Delta\setminus Z$ is reached by finitely many overlapping small discs
of this kind. Completeness makes each successive bounded neighborhood
compact. Arzel\`a--Ascoli, local holomorphic coordinates, and diagonal
extraction give a holomorphic limit $f:\Delta\setminus Z\to M$. Its
area is finite by lower semicontinuity, and $dF$ remains uniformly
bounded along it.
\emph{Step 4: puncture removal.}
Fix a puncture of this limit and move its domain coordinate to zero.
The bundle $E=f^*T^{1,0}M$ has Griffiths nonnegative curvature,
$A=dF:E\to\C^N$ is bounded and injective, and the holomorphic section
$a=f'$ has finite $L^2$ norm. Lemma~\ref{lem:extra-integrability}
applies. Fubini then gives an angle $\theta$ with
\[
\int_0^{r_0}|a(re^{i\theta})|_g^2r^{1-\sigma}\,dr<\infty.
\]
Since $\int_0^{r_0}r^{\sigma-1}\,dr<\infty$, the corresponding radial
path has finite length and converges to some $p\in M$. The bounded map
$F\circ f$ extends holomorphically to zero and its value there is
$F(p)$. Choose a relatively compact coordinate neighborhood $U$ of $p$
with compact boundary. Injectivity of $F$ gives
$\dist_{\mathrm{eucl}}(F(p),F(\partial U))>0$. For small $r$,
$F\circ f$ on $0<|z|<r$ avoids $F(\partial U)$, and the radial path
supplies a point with image in $U$. Connectedness traps the entire
punctured image in $U$. Bounded coordinate functions extend, so $f$
extends holomorphically. This uses compactness of a local chart
boundary and global injectivity of $F$; it does not require $F$ proper.
Off the puncture, $\psi_\infty=\psi\circ f$ almost everywhere; the
latter now extends smoothly across it, so the two functions define the
same distribution there and $\mu$ has no atom at the puncture. Every
alleged point of $Z$ is removable with zero atomic mass. The exponential
integrability and Morrey estimates apply across it, and the original
sequence converges on all compacts of $\Delta$. Since this applies to
every sequence, the family of centers is relatively compact.
\emph{Step 5: return to the base manifold.}
If complete images could escape every compact, choose an escaping
interior point on each disc and precompose with a disc automorphism
sending zero to that point. Both the boundary compact and the area
bound are unchanged, contradicting the center conclusion. This proves
the first assertion.
For the cotangent assertion take global holomorphic fields
$X_1,\ldots,X_m$ spanning $T^{1,0}M$, supplied by
Proposition~\ref{prop:initial-data}. The base images are compact by the
first part. The functions $\xi(X_i)$ along each disc are bounded by
their boundary values. Over the resulting compact base, the evaluation
map $\xi\mapsto(\xi(X_i))_i$ has a uniform positive least singular
value. Bounded values therefore bound the cotangent vectors and put the
images in a compact subset of $Y$. Local holomorphic normal-family
compactness gives the stated subsequence conclusion.
\end{proof}
\subsection{Boundary factorization and holomorphic frames}
\label{sec:discs:frames}
We now build frames normalized on the boundary of each disc. Their
symplectic coefficients and their values at a fixed compact set of
centers must remain controlled when the disc area increases.
Fix the injective immersion
\begin{equation}\label{disc:immersion-y}
G:Y\longrightarrow\C^q,\qquad
G(x,\xi)=\bigl(F(x),\xi(X_1(x)),\ldots,\xi(X_m(x))\bigr).
\end{equation}
It is injective because its base entries separate base points and the
fiber evaluations separate covectors. It is immersive because $dF$
detects base tangent vectors and the evaluations detect vertical ones.
There is also a finite family of global holomorphic fields spanning
$TY$: take the cotangent lifts
\[
\widetilde X_i=\sum_aX_i^a\partial_{x_a}
-\sum_{a,b}\xi_b(\partial_{x_a}X_i^b)\partial_{\xi_a}
\]
and the vertical translations by $dF_j$. The former span after
projection to $TM$, the latter span the vertical tangent. Smooth
coefficients on any fixed compact subset of $Y$ can be represented by
smooth functions of $G$ there, via local embedded coordinate
neighborhoods and a finite partition of unity.
For vector or matrix functions $a$ on the circle, write
\begin{equation}\label{disc:half-sobolev}
D(a)^2=\sum_{k\in\mathbb Z}|k|\,\|a_k\|_F^2
=\frac1{(2\pi)^2}\int_0^{2\pi}\!\!\int_0^{2\pi}
\frac{\|a(e^{i(t+\theta)})-a(e^{it})\|_F^2}
{|e^{i\theta}-1|^2}\,dt\,d\theta,
\end{equation}
where $a_k$ are Fourier coefficients and $\|\cdot\|_F$ the Frobenius
norm. For holomorphic $a$, this is
$\pi^{-1}\int_\Delta\|a'(z)\|_F^2\,dA$.
\begin{lemma}[Disc frames and Gauss area]\label{lem:disc-frames}
Let $f$ be a holomorphic disc in $Y$ extending past the unit circle.
There is a holomorphic frame $E_f$ of $f^*TY$, smooth on the closed
disc, such that $U=dG\,E_f$ satisfies
\begin{equation}\label{disc:frame-unitary}
U^*U=I\quad\text{on }\partial\Delta,\qquad
\sup_\Delta\|U\|_{\mathrm{op}}\le1.
\end{equation}
Let $\mathfrak g:Y\to\Gr(2n,q)$ send $y\mapsto dG(T_yY)$, and normalize
the Pl\"ucker form by
$\omega_{\Gr}=i\partial\bar\partial\log\det(V^*V)$ for a local
holomorphic frame $V$ of its tautological plane. Then
\begin{equation}\label{disc:gauss-area}
D(U)^2=\frac1{2\pi}\int_\Delta(\mathfrak g\circ f)^*\omega_{\Gr}.
\end{equation}
If $f(\partial\Delta)\subset K$ for a fixed compact $K\subset Y$, the
holomorphic component matrix $J_{E_f}$ of $\mathcal J$ in this frame is
bounded throughout $\Delta$ by a constant depending only on $K$ and
$G$. If also $f(0)$ ranges in a compact $K_0\subset Y$ contained in a
fixed coordinate chart, the frame matrix at zero and its inverse are
bounded by constants depending only on $K,K_0,G$ and that chart. These
constants do not depend on the disc area or on an interior compact
containing its image.
\end{lemma}
We first trivialize $f^*TY$ and then normalize the boundary Gram matrix.
For normalization we use the smooth Wiener--Masani factorization
\cite{WienerMasani1957} in the Hardy-space form described by
Berndtsson--Rosay \cite[Theorem~2.1, Section~6]{BerndtssonRosay}. We
include the construction and the boundary half-Sobolev estimates that
will control the later deformation error.
\begin{lemma}[Boundary factorization]\label{lem:boundary-factorization}
Let $L$ be an $r\times r$ smooth Hermitian matrix on the circle with
$mI\le L\le MI$, $0<m\le M<\infty$. There is a holomorphic invertible
$H$ on $\Delta$, smooth with smooth inverse on $\overline\Delta$, such
that $L=H^*H$ on the circle. It satisfies
\begin{align}
\|H\|_{\infty,\mathrm{op}}&\le\sqrt M,&
\|H^{-1}\|_{\infty,\mathrm{op}}&\le m^{-1/2},\label{disc:factor-sup}\\
D(H)^2&\le\frac M{2m^2}D(L)^2,&
D(H^{-1})^2&\le\frac1{2m^3}D(L)^2.
\label{disc:factor-seminorm}
\end{align}
The seminorm constants depend only on the boundary spectral bounds,
independently of higher derivatives of $L$.
\end{lemma}
\begin{proof}
Equip the vector Hardy space with $\|v\|_L^2=\langle v^*Lv\rangle$,
brackets denoting normalized circle average; this norm is equivalent to
the ordinary Hardy norm. The shift $v\mapsto zv$ is an isometry, its
range has codimension $r$, and $\bigcap_{k\ge0}z^kH^2=\{0\}$. An
orthonormal basis $p_1,\ldots,p_r$ of the orthogonal complement of the
range, followed by all its successive shifts, is therefore an
orthonormal basis of the weighted space, since the orthogonal remainder
after the first $k$ shift levels is $z^kH^2$, whose intersection is
zero.
Let $P$ be the holomorphic matrix with columns $p_i$. Orthogonality of
all their shifts says every Fourier coefficient of $P^*LP-I$ vanishes,
so $P^*LP=I$ almost everywhere on the circle. Thus $P$ is bounded.
Multiplication by $P$ maps the ordinary Hardy space isometrically onto
the weighted space by the basis property. Applying this to constant
target columns gives a holomorphic $Q$ of Hardy functions with $PQ=I$.
Consequently $P$ is invertible everywhere, $Q=P^{-1}$, and the boundary
identity gives $L=Q^*Q$. It also bounds $P$ and $Q$ in $H^\infty$ by
their boundary values. Taking $H=Q$ proves \eqref{disc:factor-sup}.
\emph{Seminorm.} Put $L_\theta(z)=L(e^{i\theta}z)$ and
$A_\theta(z)=H(e^{i\theta}z)H(z)^{-1}$. This is holomorphic with
$A_\theta(0)=I$, hence mean $I$, and
\[
E_\theta:=\langle\|A_\theta-I\|_F^2\rangle
=\langle\tr(L_\theta L^{-1}-I)\rangle.
\]
Changing variables in the expression for $E_{-\theta}$ gives
\begin{align*}
E_\theta+E_{-\theta}
&=\bigl\langle\tr\bigl((L_\theta-L)L^{-1}
(L_\theta-L)L_\theta^{-1}\bigr)\bigr\rangle\\
&\le m^{-2}\langle\|L_\theta-L\|_F^2\rangle.
\end{align*}
The identities
\[
H_\theta-H=(A_\theta-I)H,\qquad
H_\theta^{-1}-H^{-1}=H_\theta^{-1}(I-A_\theta)
\]
bound their squared differences by $ME_\theta$ and $m^{-1}E_\theta$.
Integrating against the symmetric kernel in \eqref{disc:half-sobolev}
proves \eqref{disc:factor-seminorm}.
\emph{Smoothness.} The same difference inequality, divided by
$\theta^2$, first gives one full $L^2$ tangential derivative of $H$ and
$H^{-1}$. With prime denoting that derivative, set $B=H'H^{-1}$; it is
the boundary value of $izH_zH^{-1}$ with zero constant Fourier
coefficient. Differentiating $L=H^*H$ gives
\[
H^{-*}L'H^{-1}=B+B^*.
\]
The strictly positive Fourier projection recovers $B$. The
one-dimensional Sobolev algebra property now bootstraps: if
$H,H^{-1}\in W^{k,2}$, the displayed expression is in $W^{k,2}$, hence
so is $B$, and $H'=BH$ gives $H\in W^{k+1,2}$; the inverse derivative
formula gives the same for $H^{-1}$. For this algebra assertion,
Cauchy--Schwarz on Fourier coefficients and
$\sum_k(1+k^2)^{-1}<\infty$ give $W^{1,2}(S^1)\subset L^\infty(S^1)$.
In each Leibniz term of total order $k\ge1$, place a derivative of order
at most $k-1$ in $L^\infty$ and the other factor in $L^2$, giving the
$W^{k,2}$ product estimate. The positive-frequency projection has norm
at most one in every $W^{k,2}$. Iterating gives all Sobolev orders, and
the same Fourier estimate for derivatives proves smoothness on the
closure.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:disc-frames}]
\emph{Preliminary frame.} The bundle $E=f^*TY$ is embedded by $dG$ in a
trivial bundle and is generated by the pullbacks of the finite global
spanning fields. Start on a disc of radius greater than one and choose
a generator not identically zero. On a slightly smaller disc, divide it
by the finitely many common zero factors of its ambient components,
using at each zero the minimum component vanishing order. The resulting
$e$ is holomorphic, remains in $E$ by continuity, and is nowhere zero.
Write its ambient components as $e_1,\ldots,e_q$ and put
$\ell_j=\overline{e_j}/|e|^2$, so $\sum_je_j\ell_j=1$. The smooth
$(0,1)$-forms
\[
b_{jk}=\ell_j\dbar\ell_k-\ell_k\dbar\ell_j
\]
are antisymmetric and satisfy $\sum_je_jb_{jk}=\dbar\ell_k$. Solve
$\dbar c_{jk}=b_{jk}$ with $c_{jk}=-c_{kj}$ on a further slightly
smaller disc: multiply the coefficient of each $(0,1)$-form by one
common smooth compactly supported cutoff equal to one on that disc,
convolve on $\C$ with $1/(\pi z)$, and use
$\dbar(1/(\pi z))=\delta_0$. This does not use a frame of $E$.
The functions
\[
a_k=\ell_k-\sum_je_jc_{jk}
\]
are holomorphic and obey $\sum_ke_ka_k=1$. The ambient row $a$
restricts to a holomorphic dual of $e$ on $E$, so
$E=\C e\oplus\ker a$. If $s_i$ are the current generators, then
$s_i-e\,a(s_i)$ generate $\ker a$. So the kernel is again an ambiently
embedded, finitely generated holomorphic bundle, now of rank one less.
Repeat on a finite nested sequence of discs whose radii stay greater
than one. Induction gives a holomorphic frame of $f^*TY$ on a
neighborhood of $\overline\Delta$.
\emph{Normalization.} Apply Lemma~\ref{lem:boundary-factorization} to
the $dG$-Gram matrix of this preliminary frame on the circle; multiply
the frame by $H^{-1}$ to get $E_f$. The first identity of
\eqref{disc:frame-unitary} is immediate. For each constant column $v$,
the maximum principle applied to the holomorphic $Uv$ gives
$|Uv|\le|v|$; the second identity follows.
\emph{Gauss area identity.} Put $Q=U^*U$; since $Q=I$ on the circle,
$\partial_r\tr Q=\tr(\partial_rQ)=\partial_r\log\det Q$ there.
Green's formula and holomorphicity give
\[
4\pi D(U)^2
=\int_\Delta\Delta_{\mathrm{eucl}}\tr Q\,dA
=\int_\Delta\Delta_{\mathrm{eucl}}\log\det Q\,dA
=2\int_\Delta(\mathfrak g\circ f)^*\omega_{\Gr},
\]
proving \eqref{disc:gauss-area}.
\emph{Bounds on $\mathcal J$.} On the boundary, $E_f$ is orthonormal
for the metric $G^*g_{\mathrm{eucl}}$, so the components of $\mathcal J$
there are bounded by its norm in that metric on $K$. They are
holomorphic functions in the frame, so the maximum principle gives the
same bound inside.
\emph{Bounds at the center.} In the fixed chart at the center write
$E_f=\partial S$ and let $J$ be the invertible coordinate matrix of
$\mathcal J$. Then
\[
J_{E_f}=S^{-1}JS^{-\mathsf T},\qquad
S^{-1}=J_{E_f}S^{\mathsf T}J^{-1}.
\]
Let $\gamma>0$ be the infimum on $K_0$ of the least singular value of
$dG$ in the coordinate Euclidean metric. By \eqref{disc:frame-unitary},
$\|S(0)\|_{\mathrm{op}}\le\gamma^{-1}$. The preceding identity gives
\[
\|S(0)^{-1}\|_{\mathrm{op}}
\le\sup_\Delta\|J_{E_f}\|_{\mathrm{op}}\,
\gamma^{-1}\sup_{K_0}\|J^{-1}\|_{\mathrm{op}}.
\]
Every quantity on the right has the stated dependence.
\end{proof}
% =============================================================================
\section{A differential inequality for the disc envelope}
\label{sec:variation}
Let $v$ be the lower semicontinuous envelope defined in
\eqref{var:envelope}--\eqref{var:sandwich}, with its fixed horizon $T$.
We continue to denote physical time at a disc center by $s$. The
following inequality is the input to the fiber optimization of
Section~\ref{sec:ellipsoids}.
\begin{proposition}[Differential inequality for the envelope]
\label{prop:disc-inequality}
Suppose a smooth real function $\varphi$ touches $v$ from below at
$(y_*,s_*)$ with $0<s_*<T$. In holomorphic cotangent coordinates write
\[
W=\dd_Y\varphi(y_*,s_*)=
\begin{pmatrix}A&B\\B^*&C\end{pmatrix},\qquad C>0.
\]
There is a Hermitian form $K$ on the horizontal coordinate space such
that
\begin{equation}
K\ge0,\qquad K\ge A-BC^{-1}B^*,\qquad
\partial_s\varphi(y_*,s_*)\ge\tr(C^{\mathsf T}K).
\label{var:viscosity-inequality}
\end{equation}
The transpose is the tangent--cotangent contraction in these
coordinates. In particular the statement applies to lower tests
translated in a fixed cotangent coordinate chart; the contraction has
constant coefficients there.
\end{proposition}
The proof goes through localized almost-minimizing discs. The estimate
on their deformations is stated with its constants: the compact
containing the interiors of the discs may depend on the area penalty,
whereas the coefficient multiplying that penalty depends only on
boundary data. After the deformation estimate, we remove the negative
part of the boundary Hessian and factor the remaining positive form. A
backward clock variation bounds its symplectic cost. The final
Schur-complement limit gives \eqref{var:viscosity-inequality}.
\subsection{Localization and order of approximation}
\label{sec:variation:localization}
Subtracting a positive multiple of $|y-y_*|^4+|s-s_*|^4$ from
$\varphi$ makes the contact strict on a compact coordinate cylinder
$\mathcal C\Subset Y\times(0,T)$ without changing the derivatives in
\eqref{var:viscosity-inequality}. Thus $v-\varphi\ge0$ on
$\mathcal C$, with equality only at $(y_*,s_*)$, and with a positive
gap near its boundary. The definition of lower semicontinuous
regularization gives
\[
\inf_{(y,s)\in\mathcal C}\bigl(v^{\mathrm{raw}}(y,s)-\varphi(y,s)\bigr)=0.
\]
On an interval slightly larger than $[0,T]$, take $Q$ and $\rho_j$ from
Theorem~\ref{thm:localized-flow} and
Proposition~\ref{prop:scalar-comparison}, and set
\begin{equation}
d_{0,j}(x,t)=j^{-1}Q(x,t)+\rho_j(x,t).
\label{var:base-cost}
\end{equation}
Thus $d_{0,j}\ge0$, $(\partial_t-\Delta_{h_t})d_{0,j}\ge0$, and
$d_{0,j}\to0$ uniformly on compact space--time sets. Choose increasing
$A_j\to\infty$ so that
\begin{equation}
d_{0,j}(x,t)\ge\rho(x,t)+j\quad\hbox{if }u(x)>A_j/3.
\label{var:tail-wall}
\end{equation}
This is possible because $\rho_j\ge\rho-C_j$ and $Q$ has a positive
quadratic lower bound. Choose fixed smooth scalar profiles and set
$\delta_1=\chi(u/A_j)$ and
$\delta_2=\widetilde\chi(N(\cdot,0)/B_j^2)$, with $0\le\delta_i\le1$,
where $\delta_1>0$ exactly when $u<A_j$, and $\delta_1=1$ when
$u\le2A_j/3$, while $\delta_2>0$ exactly when $N(y,0)<B_j^2$ and
$\delta_2=1$ when $N(y,0)\le B_j^2/2$. Here $B_j\ge j$ will be
increased later. Smooth cutoffs flat at their zero sets are
permissible. On the open boundary region put
\begin{equation}
k_i=\delta_i^{-1}-1,\qquad
d(y,t,s)=d_{0,j}(x,t)+(1+s)(k_1(x)+k_2(y)).
\label{var:wall-cost}
\end{equation}
Only the boundary of a competing disc is subject to these spatial
restrictions, whose closed boundary region is a compact
$K_j\subset Y$. Each fixed $d_{0,j}$ is smooth one-sided at $t=0$,
with all derivatives bounded on compact sets. When needed, use a smooth
extension matching these initial jets; extending it by zero to negative
time is unnecessary.
Use the boundary seminorm of Section~\ref{sec:discs},
\[
D(a)^2=\sum_{k\in\mathbb Z}|k|\,\|a_k\|^2,
\]
and add the nonnegative energy
\begin{equation}
\begin{aligned}
\mathcal E(f,w)={}&\int_\Delta\bigl(x_f^*\omega_g+
f^*G^*\omega_{\mathrm{eucl}}+
(\mathfrak g\circ f)^*\omega_{\Gr}+w^*\omega_{\mathrm{eucl}}\bigr)\\
&+D\bigl(\log(\delta_1\delta_2)\circ f\bigr)^2.
\end{aligned}
\label{var:energy}
\end{equation}
Here $x_f=\pi\circ f$, and $\mathfrak g$ is the Gauss map of the
immersion $G$ from Lemma~\ref{lem:disc-frames}. Let $\mu_{j,\eps}$ be
the infimum, over centers in $\mathcal C$ and discs with the stated
boundary restrictions, of
\begin{equation}
\mathcal F_{j,\eps}(f,w,s)=
\langle N(f,t)+d(f,t,s)\rangle+\eps\mathcal E(f,w)
-\varphi(f(0),s),\qquad t=s e^{w+\bar w}.
\label{var:penalized-functional}
\end{equation}
Infima suffice; no minimizing disc is asserted to exist. Every fixed
admissible smooth disc has finite energy, and every fixed disc in
\eqref{var:envelope} eventually has $\delta_1=\delta_2=1$ on its
boundary. Consequently
\begin{equation}
\mu_{j,0}\longrightarrow0,\qquad
\mu_{j,\eps}\downarrow\mu_{j,0}\quad(\eps\downarrow0).
\label{var:infimum-limits}
\end{equation}
For a minimizing sequence indexed by $\nu$, at fixed $j,\eps>0$,
nonnegativity of the unpenalized excess gives
\begin{equation}
\limsup_\nu\eps\mathcal E_\nu\le\mu_{j,\eps}-\mu_{j,0},\qquad
\lim_{\eps\downarrow0}\limsup_\nu\eps(1+\mathcal E_\nu)=0.
\label{var:vanishing-penalty}
\end{equation}
As $j\to\infty$ after these limits, the centers approach $(y_*,s_*)$
and
\begin{equation}
\langle d\rangle\longrightarrow0,\qquad
\langle N(f,t)\rangle\le C.
\label{var:small-walls}
\end{equation}
Indeed each of $v-\varphi$, the boundary cost $d$, and $\eps\mathcal E$
is nonnegative and bounded by \eqref{var:penalized-functional}. The
strict contact and the boundary gap in $\mathcal C$ imply the assertion
about centers, which therefore have a uniform interior margin for large
$j$, small $\eps$, and late terms of the sequences.
At fixed $j,\eps$ the energies are bounded.
Proposition~\ref{prop:disc-compactness} then confines all images of $f$
to a common compact $K'_{j,\eps}$. There is no claim that $K'_{j,\eps}$
is independent of $\eps$. One may require the initial discs to extend
holomorphically past the circle: replacing $(f,w)$ by
$(f(r\zeta),w(r\zeta))$ with $r\uparrow1$ preserves their centers and
converges in every boundary norm used here, and strict boundary
inequalities persist for $r$ close to one. Choosing the approximation
separately for each term preserves all infima and minimizing-sequence
assertions above.
\subsection{Boundary calculus and holomorphic deformations}
\label{sec:variation:deformation}
We first record the boundary estimates that distinguish the two kinds
of constants. The difference-quotient formula for $D$ gives
\begin{equation}
\begin{aligned}
D(ab)&\le\|a\|_\infty D(b)+\|b\|_\infty D(a),\\
D(\Psi(a))&\le\Lip(\Psi)D(a),\\
|\langle a,\partial_\theta b\rangle|&\le D(a)D(b).
\end{aligned}
\label{var:boundary-calculus}
\end{equation}
The last inequality follows directly from Fourier series; it is the
pairing between $\dot H^{1/2}$ and $\dot H^{-1/2}$. All three formulas
hold componentwise for finite-dimensional vectors and matrices, with
fixed dimensional constants. Smooth functions of bounded data are
composed and multiplied with constants determined by their smooth
bounds on the relevant compact sets. The circular Hilbert transform
preserves $D$ on mean-zero functions.
For a disc let $E_f$ be the frame in Lemma~\ref{lem:disc-frames}, and
put
\[
X=G\circ f,\qquad U=dG\,E_f,\qquad p=(\delta_1\delta_2)\circ f,
\qquad q=\log p.
\]
In the estimates below, $p,q$ are these scalar functions; the symplectic
polynomial is distinguished by its matrix argument. For an integer
$a\ge4$, to be enlarged once below, let $m=p^a$ and let $h$ be the
scalar outer function with boundary modulus $m$ and $h(0)>0$. Thus
\begin{equation}
h(0)=\exp\langle\log m\rangle,\quad \|h\|_\infty\le1,
\quad D(h)+D(h/m)\le C_aD(q).
\label{var:outer-wall}
\end{equation}
Here $h/m$ denotes only its boundary phase. To verify the last bound,
write that phase as $\exp(i\mathcal H(aq))$; both the exponential on the
imaginary axis and $q\mapsto e^{aq}$ for $q\le0$ are Lipschitz. The
frame and energy estimates give
\begin{equation}
\|U\|_\infty\le1,\qquad
D(X)+D(U)+D(w)+D(q)\le C(1+\sqrt{\mathcal E}).
\label{var:basic-boundary-bounds}
\end{equation}
Constants absorbing fixed area normalizations are harmless. Since
$t\le T$, restricting the exponential to the range of $2\Re w$ gives
$D(t)\le C_TD(w)$, even if some boundary times approach zero.
For $t$ and $s$ held fixed in the spatial Hessian, define
\begin{equation}
W_b=\dd_Y(N+d),\qquad T_b=m^2E_f^*W_bE_f.
\label{var:boundary-symbol}
\end{equation}
For large enough fixed $a$,
\begin{equation}
\|T_b\|_\infty\le C_j,\qquad
D(T_b)^2\le C_j(1+\mathcal E).
\label{var:symbol-bounds}
\end{equation}
For example $\dd(\delta^{-1})=2\delta^{-3}\partial\delta\otimes\bar\partial\delta
-\delta^{-2}\dd\delta$. Multiplication by $p^{2a}$ clears all its
denominators and leaves smooth coefficients on the closed boundary
region. Every other coefficient is smooth on
$K_j\times[0,T]\times\mathcal C_s$. Choose $K_j\subset\mathcal W_j
\Subset Y$ with compact closure. The injective immersion $G$ restricted
to $\overline{\mathcal W_j}$ has a continuous inverse onto its image by
compactness, so $G|_{\mathcal W_j}$ is an embedding. A finite extension
and partition construction from $G(\mathcal W_j)$ near $G(K_j)$
expresses these coefficients as bounded smooth functions of $X$
there. The same applies to tensor coefficients evaluated on $U$.
Equation~\eqref{var:symbol-bounds} follows from
\eqref{var:boundary-calculus} and \eqref{var:basic-boundary-bounds}.
These coefficient extensions are used only along the embedded compact
set; no properness of $G$ is asserted.
\begin{lemma}[Deformation estimate, with its uniformities]
\label{lem:disc-deformation}
Use the preceding localization, with $j$ sufficiently large and $\eps>0$
sufficiently small that the minimizing-sequence centers have a common
interior margin in $\mathcal C$. Fix a bound $L<\infty$ and such a
sequence for \eqref{var:penalized-functional}, with energy bounded by
$E_0<\infty$. Let $l_\nu$ be any holomorphic columns, smooth on the
closed disc, with
\[
\|l_\nu\|_\infty\le L,\qquad D(l_\nu)\le D_0<\infty.
\]
They may depend on the disc. With
$\varphi_{E,c}=E_f(0)^*\dd_Y\varphi(f(0),s)E_f(0)$, one has
\begin{equation}
\limsup_{\nu\to\infty}
\left\{
h(0)^2\varphi_{E,c}(l(0),\overline{l(0)})
-\langle T_b(l,\bar l)\rangle
-C_j(L)\eps\bigl(1+\mathcal E+D(l)^2\bigr)
\right\}\le0.
\label{var:deformation-estimate}
\end{equation}
The constant $C_j(L)$ depends only on $L$, the dimension, the chosen
integer $a$, the fixed time interval and center cylinder, and smooth
coefficient bounds for the metrics, immersion, Gauss map and cutoffs on
a fixed neighborhood of the boundary compact $K_j$. It is independent
of $\eps,E_0,D_0$, of the interior compact $K'_{j,\eps}$, and of the
auxiliary loss introduced in the proof. In contrast, the existence
radius of the deformations, bounds on their full coefficients, and
bounds on Taylor remainders may depend on $j$, $L$, $E_0$, $D_0$,
$K'_{j,\eps}$, that loss, the interior margin of the centers, and the
fixed smooth test $\varphi$.
\end{lemma}
\begin{proof}
We give the realization, Taylor justification, and actual
mixed-derivative estimate separately.
\emph{Realization of the first field.}
Let $Z_1,\ldots,Z_b$ be the global holomorphic spanning fields on $Y$.
In the immersion $G$ let $A$ be their column matrix along $f$. If
$r=\dim_\C Y$, enumerate all $r\times r$ minors $a_i$ of $A$; on
$K'_{j,\eps}$ the sum of their squared moduli has a common positive
lower bound. Their sup and Dirichlet norms are bounded at fixed
parameters: the minors are smooth holomorphic functions of $X$ on this
compact, and $D(X)$ is bounded. Set
\[
\sigma_i=\frac{\overline{a_i}}{\sum_k|a_k|^2},\qquad
b_{ik}=\sigma_i\dbar\sigma_k-\sigma_k\dbar\sigma_i.
\]
These $(0,1)$ coefficients have bounded $L^2(\Delta)$ norm. Solve
$\dbar Q_{ik}=b_{ik}$, skew-symmetrically, with bounded $W^{1,2}$ norm.
Explicitly, if $b_{ik}=b_{ik,0}\,d\bar\zeta$, take
$Q_{ik}=4\partial_\zeta\Delta_D^{-1}b_{ik,0}$, with the convention
$\Delta=4\partial_\zeta\partial_{\bar\zeta}$. Here $\Delta_D^{-1}$ is
the zero-Dirichlet inverse on the unit disc. The unit disc has smooth
boundary, the principal coefficients are the identity, all lower
coefficients vanish, so the scalar Dirichlet estimate gives
\[
\|\Delta_D^{-1}b_{ik,0}\|_{W^{2,2}(\Delta)}
\le C\|b_{ik,0}\|_{L^2(\Delta)}
\]
\cite[Theorem~9.15, Lemma~9.17]{GilbargTrudinger}, proving the asserted
$W^{1,2}$ bound. Linearity preserves skew symmetry, and smooth input
gives smooth output \cite[Theorem~8.13]{GilbargTrudinger}.
\emph{Boundary seminorm.} For a smooth $q$ on the closed disc with
boundary Fourier coefficients $a_k$, let
$h(re^{i\theta})=\sum_ka_kr^{|k|}e^{ik\theta}$ be its harmonic
extension. Integration by parts gives
\[
\int_\Delta|\nabla q|^2
=\int_\Delta|\nabla h|^2+\int_\Delta|\nabla(q-h)|^2
\ge2\pi\sum_k|k||a_k|^2.
\]
If $q_0(r)$ is the angular average, then for $1/2\le r\le1$,
\[
|q_0(1)|^2\le2|q_0(r)|^2+2(1-r)\int_r^1|q_0'(s)|^2\,ds.
\]
Integrating in $r$ and applying Jensen bounds the constant coefficient
$|a_0|^2$ by $C\|q\|_{W^{1,2}(\Delta)}^2$. The nonconstant $L^2$
Fourier sum is bounded by its $|k|$-weighted sum. Consequently
\[
\|q|_{\partial\Delta}\|_{L^2}^2+D(q|_{\partial\Delta})^2
\le C\|q\|_{W^{1,2}(\Delta)}^2.
\]
Applied entrywise to $Q_{ik}$, this bounds boundary $D$ and $L^2$
norms. The operator constants are fixed-disc constants; the input
norms can still depend on $K'_{j,\eps}$. The functions
\[
g_i=\sigma_i+\sum_kQ_{ik}a_k
\]
are holomorphic and $\sum_ig_ia_i=1$: differentiating uses
$\sum_ka_k\sigma_k=1$ and $\sum_ka_k\dbar\sigma_k=0$, and skew
symmetry cancels the quadratic term.
\emph{Normalizing the outer function.} A $W^{1,2}$ trace need not be
bounded. For a large number $P$ let $\alpha$ be the scalar outer
function with positive center value and boundary modulus
\[
r_P=(1+|Q|^2/P)^{-1}.
\]
All norms and products involving $Q$ in the boundary estimates are of
its trace on the circle. The maps $Q\mapsto\log(1+|Q|^2/P)$ have
Lipschitz constant $C/\sqrt P$ and $L^2$ norms at most
$C\|Q\|_2/\sqrt P$. The maps $Q\mapsto r_PQ$ have bounded Lipschitz
constant and sup norm at most $C\sqrt P$. The phase of $\alpha$ is the
Hilbert transform of $\log r_P$. Thus
\[
\|\alpha\|_\infty\le1,\quad
D(\alpha)\le CD(Q)/\sqrt P,\quad
D(\alpha Q)\le CD(Q),\quad
\|\alpha Q\|_\infty\le C\sqrt P.
\]
In the third estimate the $\sqrt P$ from the bounded product multiplies
the $P^{-1/2}$ bound for the phase. Moreover
\[
\langle1-|\alpha|^2\rangle\le C\|Q\|_2^2/P,
\qquad -\log\alpha(0)=\langle\log(1+|Q|^2/P)\rangle\longrightarrow0.
\]
Given $\eta>0$, choose $P$ uniformly for the sequence so that these
last two quantities are at most $\eta$ and $D(\alpha)\le1$. The
boundary estimates bound sup norms and $D$ norms of $\alpha g_i$.
These functions are holomorphic, so the maximum principle also bounds
their interior sup norms at the fixed parameters; it is not applied to
the nonholomorphic products $\alpha Q$.
\emph{The column field.} For each minor $a_i$, Cramer's rule gives a
vector $c_i$, holomorphic in the disc, with $Ac_i=a_iUl$. Use the
adjugate on its selected rows and columns and set the other coefficient
entries to zero. The identity follows by expanding the augmented
determinants: every $(r+1)$-minor of $(A,Ul)$ is zero because this
matrix still has rank $r$. This also treats selected minors that vanish
identically. Consequently $c=\sum_i\alpha g_ic_i$ has bounded sup norm
and $D$ and obeys $Ac=\alpha Ul$. Write $c=(c_1,\ldots,c_b)$ and
compose the complex flows of $Z_1,\ldots,Z_b$ with times
$zhc_1,\ldots,zhc_b$, starting at $f$. For a sufficiently small common
complex parameter $z$ this gives a holomorphic family $f_z$, smooth up
to the circle, whose first field is
\begin{equation}
V=\left.\partial_zf_z\right|_0=\alpha hE_fl.
\label{var:first-field}
\end{equation}
The existence radius is uniform at fixed parameters, since initial
images lie in $K'_{j,\eps}$ and all flow times are bounded. On the
boundary the displacement is $m$ times a uniformly bounded smooth
function. Because $m\le\delta_i^a$, decreasing this radius makes each
new $\delta_i$ comparable to its old value, with constants in
$[1/2,3/2]$. The spatial boundary restrictions persist. The variable
$w$ and all times $t$ are unchanged, so no uniform margin below the
upper time boundary is needed. The center stays in $\mathcal C$ by its
established margin.
\emph{Uniform Taylor remainders.} On the controlled interior compact the
flow map and all its derivatives through any fixed finite order are
bounded. Differentiating in the disc variable introduces one
derivative of $f$, $h$, or $c$. Their Dirichlet norms are bounded; for
$f$ use local metric equivalence on $K'_{j,\eps}$ and
\eqref{var:energy}. The family and its first three real parameter
derivatives have bounded $W^{1,2}$ norms, and bounded sup norms for
their values. Differentiating area integrands through order three
yields only bounded coefficients times products of at most two such
first disc derivatives; Cauchy--Schwarz supplies uniform integrable
bounds. The same reasoning applies to the Gauss map, whose derivatives
are bounded on the controlled compact.
\emph{Boundary remainders.} Write $\beta=h/m$. The boundary values of
$f_z$ are a smooth function of $X$, $c$, $\beta$, and $zm$, on a fixed
compact range. More precisely, for either cutoff,
\[
\delta_i(f_z)=\delta_i(f)+mR_i(z,X,c,\beta),\qquad R_i(0,\cdot)=0,
\]
where real parameter derivatives of $R_i$ through order three have
bounded sup and $D$ norms at fixed parameters, by
\eqref{var:boundary-calculus}. Since
$m/\delta_i=\delta_i^{a-1}\delta_{3-i}^{a}$ is smooth on the closed
boundary region, the function
\[
\log\delta_i(f_z)-\log\delta_i(f)=\log\bigl(1+(m/\delta_i)R_i\bigr)
\]
and its parameter derivatives through order three have uniformly
bounded $D$ norms. For reciprocal walls
\[
\frac1{\delta_i(f_z)}-\frac1{\delta_i(f)}
=-\frac{m}{\delta_i^2}\frac{R_i}{1+(m/\delta_i)R_i},
\]
and the prefactor is smooth and bounded when $a\ge2$. Hence all nonzero
parameter derivatives of the wall costs through order three are
uniformly bounded, even when the unchanged wall cost is large. The
remaining boundary integrands $N,d_{0,j}$ have bounded derivatives on
$K_j\times[0,T]$. Finally the quadratic seminorm of the logarithmic
wall has three bounded parameter derivatives by its Hilbert-space
pairing formula. This proves a uniform $O(|z|^3)$ remainder after
parameter-circle averaging of the entire functional minus the test. The
constants here may depend on the interior compact and on $\eta$.
\emph{Actual mixed energy derivative.} Set
$\widehat V=dG\,V=\alpha hUl$ on the circle.
Equations~\eqref{var:boundary-calculus}--\eqref{var:first-field} imply
\begin{equation}
\|\widehat V\|_\infty\le L,\qquad
D(\widehat V)^2\le C_j(L)\bigl(1+\mathcal E+D(l)^2\bigr).
\label{var:first-field-bound}
\end{equation}
Here only $\|\alpha\|_\infty\le1$ and $D(\alpha)\le1$ were used; the
corona coefficients have disappeared.
\emph{Area identity.} Consider one of the closed $(1,1)$-forms $\omega$
in the three spatial area terms of \eqref{var:energy}. It is a smooth
closed form on $Y$ after pullback. In ambient coordinates near the
boundary write its intrinsic tensor as
$ib_{p\bar q}(X)dX^p\wedge d\bar X^q$, extending the coefficients
smoothly from $G(\mathcal W_j)$ near $G(K_j)$ as above. Deformed
boundaries stay intrinsically in $\mathcal W_j$ for a smaller parameter
radius. Closure is used intrinsically before this extension. Cartan's
formula and Stokes give
\begin{equation}
\begin{aligned}
\left.\partial_z\partial_{\bar z}\int_\Delta f_z^*\omega\right|_0
=i\int_0^{2\pi}\bigl[&
b_{p\bar q}(X)\widehat V^p\partial_\theta\overline{\widehat V^q}\\
&+(\partial_{\bar r}b_{p\bar q})(X)
\widehat V^p\overline{\widehat V^r}\partial_\theta\overline{X^q}\bigr]\,d\theta.
\end{aligned}
\label{var:boundary-area-identity}
\end{equation}
One can also verify the identity using local potentials: the mixed
parameter derivative equals the boundary normal derivative of
$\omega(V,\bar V)$, with the corresponding Stokes normalization. The
holomorphic dependence of $Gf_z$ eliminates a mixed second parameter
field. The identity is global; no partition of $\omega$ into nonclosed
forms is required.
With $B=\|\widehat V\|_\infty$ and bounded $C^2$ norm of $b$ at $K_j$,
\[
\begin{aligned}
D(b(X)\widehat V)&\le C_j\bigl(D(\widehat V)+BD(X)\bigr),\\
D((\dbar b)(X)\widehat V\overline{\widehat V})
&\le C_j\bigl(BD(\widehat V)+B^2D(X)\bigr).
\end{aligned}
\]
Pairing each angular derivative by \eqref{var:boundary-calculus}
bounds \eqref{var:boundary-area-identity} by
$C_j(L)(D(X)^2+D(\widehat V)^2)$. This proves the estimate for all
spatial areas. The $w$ area is unchanged. For the Euclidean area the
identity reduces, up to its fixed normalization, to $D(\widehat V)^2$.
\emph{Logarithmic seminorm.} Put $Z=\alpha\beta Ul$ so that
$\widehat V=mZ$ on the boundary. Then
\[
\|Z\|_\infty\le L,\qquad
D(Z)\le C_j(L)(1+\sqrt{\mathcal E}+D(l)).
\]
Regarding $p=\delta_1\delta_2$ as a smooth ambient function, the tensors
\begin{equation}
\begin{aligned}
m\partial\log p&=p^{a-1}\partial p,\\
m^2\dd\log p&=p^{2a-1}\dd p-p^{2a-2}\partial p\otimes\bar\partial p
\end{aligned}
\label{var:cleared-log-tensors}
\end{equation}
extend smoothly to the closed boundary region. For $q_z=\log p(f_z)$
their contractions against $Z$ and $(Z,\bar Z)$ are exactly
$\partial_zq_z|_0$ and $\partial_z\partial_{\bar z}q_z|_0$. Each has $D$
bounded by $C_j(L)(1+\sqrt{\mathcal E}+D(l))$. Write
$b=\partial_zq_z|_0$, $c=\partial_z\partial_{\bar z}q_z|_0$, and let
$B_D$ be the sesquilinear pairing
$B_D(u,v)=\sum_k|k|u_k\overline{v_k}$. Since $q_z$ is real,
differentiating this quadratic seminorm gives exactly
\begin{equation}
\left.\partial_z\partial_{\bar z}D(q_z)^2\right|_0
=2D(b)^2+2\Re B_D(q,c).
\label{var:log-energy-identity}
\end{equation}
Since $D(q)^2\le\mathcal E$, Cauchy--Schwarz proves
\begin{equation}
\left|\left.\partial_z\partial_{\bar z}\mathcal E(f_z,w)\right|_0\right|
\le C_j(L)(1+\mathcal E+D(l)^2).
\label{var:actual-energy-derivative}
\end{equation}
All constants here use boundary coefficients only, independently of the
realization and Taylor constants.
\emph{Testing the infimum.} Every deformed value of
\eqref{var:penalized-functional} is at least $\mu_{j,\eps}$, while the
values at $z=0$ approach it. Average over $|z|=r$, subtract the value at
zero, divide by $r^2$. First take the minimizing-sequence limit with $r$
fixed, then $r\downarrow0$. The uniform remainder just proved yields a
nonnegative lower limit for the mixed derivative of cost minus test. Its
nonpenalty terms are
\[
\langle|\alpha|^2T_b(l,\bar l)\rangle
-|\alpha(0)|^2h(0)^2\varphi_{E,c}(l(0),\overline{l(0)}).
\]
The center frames and their inverses are bounded independently of
interior area by Lemma~\ref{lem:disc-frames}. Thus the test term is
bounded, as is the boundary symbol by \eqref{var:symbol-bounds}.
Removing $\alpha$ changes these terms by at most $C_{j,L,\varphi}\eta$.
Equation~\eqref{var:actual-energy-derivative} has a coefficient
independent of $\eta$. Send $\eta\downarrow0$ after the sequence and
radius limits. This proves \eqref{var:deformation-estimate} with the
stated uniformities. In particular, none of the constants controlling
the shrinking realization radius multiplies the final $\eps\mathcal E$
error.
\end{proof}
\subsection{Removing the negative part and factoring the boundary form}
\label{sec:variation:negative-part}
The allowance for disc-dependent columns in
Lemma~\ref{lem:disc-deformation} is essential here. Let
$S=(T_b)_-$ be the positive semidefinite negative part of $T_b$. Matrix
clipping is Lipschitz in the Frobenius norm, so $\|S\|_\infty\le C_j$
and $D(S)^2\le C_j(1+\mathcal E)$. Let $S_P$ be the Fej\'er mean of
order $P$. Positivity of the Fej\'er kernel preserves the sup bound, and
its Fourier multipliers give
\begin{equation}
\|S_P-S\|_2^2\le C_j(1+\mathcal E)/P.
\label{var:fejer-error}
\end{equation}
Each column $l_i(\zeta)=\zeta^{P+1}S_P(\zeta)e_i$ is holomorphic,
vanishes at zero, and has $\|l_i\|_\infty\le C_j$,
$D(l_i)^2\le C_j(P+1)$. The frequency shift has modulus one on the
circle, so summing \eqref{var:deformation-estimate} over these columns
tests $\tr(T_bS_P^2)$. In comparison,
$\tr(T_bS^2)=-\tr(S^3)$.
For fixed $j,\eps$ discard finitely many sequence terms and let
$B_\eps$ bound $1+\mathcal E_\nu$, with $\eps B_\eps\to0$ as
$\eps\downarrow0$, as permitted by \eqref{var:vanishing-penalty}. Choose
$P=P_\eps$, fixed along this sequence, with
\[
B_\eps/P_\eps\longrightarrow0,\qquad
\eps P_\eps\longrightarrow0;
\]
for example round $\sqrt{B_\eps/\eps}$ upward. The trace error is at
most $C_j\|S_P-S\|_2$ by boundedness of both matrices.
Equations~\eqref{var:deformation-estimate} and
\eqref{var:fejer-error} then give
\begin{equation}
\lim_{\eps\downarrow0}\limsup_\nu\langle\tr((T_b)_-^3)\rangle=0,
\qquad
\lim_{\eps\downarrow0}\limsup_\nu\|(T_b)_-\|_1=0.
\label{var:negative-part}
\end{equation}
The second conclusion follows by H\"older's inequality in fixed
dimension. This argument uses both sides of the intermediate frequency
choice; a bound of the form $\eps\mathcal E=O(1)$ alone would not
suffice.
\emph{Positive shift.} Fix $\delta>0$ and apply the smooth scalar
function $\lambda\mapsto\delta+(\lambda+\sqrt{\lambda^2+\delta^2})/2$
to $T_b$, calling the result $L$. Then
\[
L\ge T_b+\delta I,\qquad L\ge\delta I,\qquad
\|L-(T_b)_+\|\le\tfrac32\delta.
\]
Functional calculus in fixed dimension gives
$D(L)\le C_{j,\delta}D(T_b)$. Lemma~\ref{lem:boundary-factorization}
gives $L=H^*H$ with $H,H^{-1}$ holomorphic and smooth on the closed
disc; this is the smooth boundary factorization of
\cite[Theorem~2.1]{BerndtssonRosay}, with the quantitative seminorm
estimate established in Section~\ref{sec:discs}. For every constant
unit column $e$,
\[
l=H^{-1}e,\qquad \|l\|_\infty\le\delta^{-1/2},\qquad
D(l)^2\le C_{j,\delta}(1+\mathcal E).
\]
These are permissible disc-dependent tests at fixed $j,\eps,\delta$.
Since $T_b\le L$, their boundary quadratic costs are at most one.
Equation~\eqref{var:vanishing-penalty} now gives
\begin{equation}
h(0)^2\varphi_{E,c}\le(1+o(1))H(0)^*H(0)
\label{var:center-majorant}
\end{equation}
as the sequence limit and then $\eps\downarrow0$ are taken at fixed
$j,\delta$. To see that this is a matrix assertion, if the largest
violation did not tend to zero, choose its unit eigenvector separately
for each disc. Those columns satisfy exactly the same sup and $D$
bounds, contradicting Lemma~\ref{lem:disc-deformation}. So no
fixed-test assumption has entered either the Fej\'er test or the
spectral-factor test.
\subsection{The backward clock and the symplectic cost}
\label{sec:variation:backward-clock}
Choose cutoffs $0\le\chi_i\le1$ which are one wherever $\delta_i$
differs, or starts to differ, from one. Require
\[
\supp\chi_1\subset\{u>A_j/3\},\qquad
\supp\chi_2\subset\{N(y,0)>c_2B_j^2\}
\]
for a fixed $c_2>0$. Before choosing $B_j$ we can choose $C'_j\ge1$
depending on the base compact, $d_{0,j}$ and the time interval, and a
fixed integer $c_3$, such that
\begin{equation}
\begin{aligned}
\bigl|p(W_b)-\partial_tN-&\Delta_{h_t}\bigl(d_{0,j}+(1+s)k_1\bigr)\bigr|\\
&\le C'_j(1+B_j)^{c_3}(\delta_1\delta_2)^{-c_3}\boldone_{\{\chi_2=1\}}.
\end{aligned}
\label{var:fiber-wall-error}
\end{equation}
Indeed Lemma~\ref{lem:symplectic-evolution} gives the exact equality
without the second wall. The remaining expression is a polynomial of
degree at most two in the Hessian entries. On a compact base, $N(y,t)$
and $N(y,0)$ are quadratic in the fiber variables, and their fixed
finite derivatives grow polynomially in those variables. Differentiating
the reciprocal cutoff twice introduces at most three inverse powers of
$\delta_2$; the analogous first-wall factors have at most three inverse
powers of $\delta_1$. Enlarging a fixed integer $c_3$ bounds all these
finitely many terms. This proves \eqref{var:fiber-wall-error},
including that $C'_j$ is chosen before $B_j$. Also
\begin{equation}
|\Delta_{h_t}k_1|\le Ce^\rho\delta_1^{-3}\boldone_{\{\chi_1=1\}}.
\label{var:base-wall-error}
\end{equation}
The controlled gradient and complex Hessian of $u$ bound $|\dd k_1|_g$
by $C\delta_1^{-3}$ uniformly in large $A_j$. Since $0<h_t\le g$, each
inverse eigenvalue relative to $g$ is at most $e^\rho$, proving
\eqref{var:base-wall-error}.
Fix integers $a,c$ sufficiently large that $4a\ge c_3+3$ and
$c>c_3+3$, retaining all earlier requirements on $a$. Put
\begin{equation}
\begin{aligned}
\gamma_0={}&m^4\exp\bigl[-\chi_1(\rho+c\log(j+2))\bigr]\\
&\mathrel{}\cdot\exp\bigl[-\chi_2(
\log C'_j+c\log(j+2)+c\log(1+B_j))\bigr],\\
\gamma={}&\gamma_0/\langle\gamma_0\rangle.
\end{aligned}
\label{var:clock-weight}
\end{equation}
Choose $B_j$ large so that $\log C'_j/B_j^2\to0$. The preceding choice
of $C'_j$ makes this noncircular. Since $-\log\delta_i\le k_i$,
\eqref{var:small-walls} controls $\langle-\log m\rangle$. On
$\supp\chi_1$, \eqref{var:tail-wall} controls both $\rho$ and $j$ by
$d_{0,j}$. On $\supp\chi_2$, the pointwise boundary inequality
$N(f,t)\ge N(f,0)$ and \eqref{var:small-walls} give
$\langle\chi_2\rangle\le C/B_j^2$. In the successive limits it follows
that
\begin{equation}
\langle-\log\gamma_0\rangle\longrightarrow0,\qquad
\langle\gamma_0\rangle\longrightarrow1,\qquad
\langle\log\gamma\rangle\longrightarrow0.
\label{var:clock-loss}
\end{equation}
For example, the second limit also follows from
$e^{\langle\log\gamma_0\rangle}\le\langle\gamma_0\rangle\le1$.
The powers and exponentials in \eqref{var:clock-weight} bound the
weighted errors in \eqref{var:fiber-wall-error} and
\eqref{var:base-wall-error} by a quantity tending to zero. On the
support of each error its cutoff is one, so the cancellation is literal;
$m^4$ clears the displayed inverse wall powers, $e^{-\rho}$ clears
$e^\rho$, and $(C'_j)^{-1}(1+B_j)^{-c}(j+2)^{-c}$ clears the fiber
error. The denominator $\langle\gamma_0\rangle$ tends to one, and
$t/s$ is bounded because centers stay in $\mathcal C$. Since
$(\partial_t-\Delta_{h_t})d_{0,j}\ge0$,
\begin{equation}
\left\langle\frac ts\gamma(\partial_tN+\partial_td_{0,j})
+k_1+k_2\right\rangle
\ge\left\langle\frac ts\gamma p(W_b)\right\rangle-o_j(1).
\label{var:clock-comparison}
\end{equation}
Here $o_j(1)$ tends to zero after the earlier limits.
\emph{Bounds on the clock test.} The clock test giving the left side of
\eqref{var:clock-comparison} runs backward. At fixed $j$,
\[
\|\gamma\|_\infty\le C_j,\qquad
D(\gamma)^2\le C_j(1+\mathcal E).
\]
These follow by \eqref{var:boundary-calculus} from the smooth
coefficients in \eqref{var:clock-weight}; the denominator has a
positive lower bound by Jensen and the bounded cost. For an explicit
bound, let $M$ bound the unpenalized boundary cost along the fixed-$j$
sequences. Then $\langle k_1+k_2\rangle\le M$,
$\langle-\log p\rangle\le M$, and
$\langle\chi_1\rho\rangle\le M$ by the tail wall. So
\[
\langle-\log\gamma_0\rangle
\le(4a+1)M+2c\log(j+2)+\log C'_j+c\log(1+B_j),
\]
and Jensen gives a positive lower bound for $\langle\gamma_0\rangle$
depending only on these fixed-$j$ data. No lower bound on the individual
cutoffs or boundary times is used.
\emph{Backward clock.} Let $\Gamma$ be holomorphic with boundary real
part $\gamma-1$ and $\Gamma(0)=0$. For $s'<s$ set
\[
w'=w+\tfrac12\log(s'/s)\Gamma.
\]
On the circle the new time is $t(s'/s)^\gamma$. In the interior its
exponent is the positive harmonic extension of $\gamma$. Hence every
time decreases, including when the original disc has arbitrarily little
margin below $T$. The center of $w'$ is still zero.
Compare the perturbed functional with its infimum, first along the
minimizing sequence and then letting $s'\uparrow s$:
\begin{equation}
\partial_s\varphi(f(0),s)
\ge\left\langle\frac ts\gamma(\partial_tN+\partial_td_{0,j})
+k_1+k_2\right\rangle
-C_j\eps(1+\mathcal E)-o(1).
\label{var:backward-test}
\end{equation}
The sign follows because the quotient denominator $s'-s<0$. For the
energy error only the $w$ area changes, and $D(\Gamma)\le CD(\gamma)$
bounds its derivative by Cauchy--Schwarz. Its Taylor expansion is
quadratic in $\log(s'/s)$. For the boundary cost, derivatives of
$t(s'/s)^\gamma$ are uniformly bounded at fixed $j$; $N,d_{0,j}$ are
smooth down to $t=0$ on $K_j$. The possibly unbounded unchanged walls
multiply $1+s'$ affinely and have no second time remainder. These facts
justify the sequence-then-difference limit in
\eqref{var:backward-test} without a bound on the imaginary part of
$\Gamma$.
\emph{Replacing by a positive form.} Let $\mathcal J_E$ be the bivector
components in $E_f$. Its holomorphic sup bound depends only on $K_j$ by
Lemma~\ref{lem:disc-frames}. Since $p$ is quadratic in the Hermitian
form, $p(W_b)=m^{-4}p(T_b,\mathcal J_E)$. At fixed $j$ the factor
$\gamma m^{-4}$ is bounded. So \eqref{var:negative-part}, the uniform
symbol bounds, and $L=(T_b)_++O(\delta)$ allow the replacement
\begin{equation}
\left\langle\frac ts\gamma p(W_b)\right\rangle
=\left\langle\frac ts\gamma m^{-4}p(L,\mathcal J_E)\right\rangle
+O_j(\delta)+o(1)
\label{var:positive-replacement}
\end{equation}
after $\eps\downarrow0$. The vector $(\bigwedge^2H)\mathcal J_E$ is
holomorphic and nonzero. Logarithmic submean for its squared norm,
followed by scalar Jensen, gives
\begin{equation}
\begin{aligned}
\left\langle\frac ts\gamma m^{-4}p(L,\mathcal J_E)\right\rangle
&\ge\exp\langle\log\gamma\rangle\,
h(0)^{-4}p(H(0)^*H(0),\mathcal J_E(0)).
\end{aligned}
\label{var:symplectic-submean}
\end{equation}
Indeed $\langle\log(t/s)\rangle=2\Re w(0)=0$ and
$\log h(0)=\langle\log m\rangle$, which give precisely the factors in
\eqref{var:symplectic-submean}. If desired, logarithmic submean follows
by applying submean to $\log(|Z|^2+\eta)$ and then $\eta\downarrow0$
for the displayed holomorphic vector $Z$.
\subsection{Passing from positive majorants to the required form}
\label{sec:variation:schur}
\begin{lemma}[Symplectic cost and the Schur complement]
\label{lem:symplectic-schur}
In a cotangent vector space with canonical bivector
$\mathcal J=\sum_ie_i\wedge f_i$, let $\mathcal H>0$ be Hermitian, and
suppose $\mathcal H\ge W$, where the vertical block $C$ of $W$ is
positive. Then
\begin{equation}
S=\Schur(\mathcal H)\ge0,\qquad
S\ge\Schur(W),\qquad
p(\mathcal H)\ge\tr(C^{\mathsf T}S).
\label{var:schur-cost}
\end{equation}
\end{lemma}
\begin{proof}
Write
$\mathcal H=\left(\begin{smallmatrix}A&B\\B^*&D\end{smallmatrix}\right)$,
$S=A-BD^{-1}B^*$, $R=D^{-1}B^*$. The complex-linear change
$(x,y)\mapsto(x,y+Rx)$ makes the form block diagonal with blocks
$(S,D)$. It carries the canonical bivector to
\[
\sum_ie_i\wedge f_i+\sum_{k,i}R_{ki}f_k\wedge f_i.
\]
Its cross and vertical components are orthogonal for the induced
exterior-square form, so
\[
p(\mathcal H)=\tr(SD^{\mathsf T})
+\Bigl|\sum_{k,i}R_{ki}f_k\wedge f_i\Bigr|_D^2.
\]
The shear need not be symplectic; its additional vertical term is
retained and nonnegative. Since $D\ge C$ and $S>0$, this gives
$p(\mathcal H)\ge\tr(SC^{\mathsf T})$. For each horizontal $x$,
\[
S(x,\bar x)=\inf_y\mathcal H((x,y),\overline{(x,y)})
\ge\inf_yW((x,y),\overline{(x,y)})
=\Schur(W)(x,\bar x).
\]
The last infimum is finite and has the usual Schur expression because
$C>0$. This proves the lemma.
\end{proof}
\begin{proof}[Completion of the proof of Proposition~\ref{prop:disc-inequality}]
Combine \eqref{var:center-majorant}, \eqref{var:clock-comparison},
\eqref{var:backward-test}, \eqref{var:positive-replacement}, and
\eqref{var:symplectic-submean}. More explicitly, if $r\ge0$ denotes the
error tending to zero in \eqref{var:center-majorant}, the form with
frame matrix
\[
\mathcal H_E=(1+r)h(0)^{-2}H(0)^*H(0)
\]
is a positive majorant of the test Hessian at its center, after an
arbitrarily small enlargement if necessary. Transform it to the fixed
cotangent coordinates. Its symplectic cost is
\[
p(\mathcal H)=(1+r)^2h(0)^{-4}p(H(0)^*H(0),\mathcal J_E(0)).
\]
For fixed $j,\delta$ first take the minimizing-sequence limits and
$\eps\downarrow0$; then send $\delta\downarrow0$; finally $j\to\infty$.
Equations~\eqref{var:small-walls} and \eqref{var:clock-loss} give a
diagonal sequence of approaching centers and positive majorants
$\mathcal H$ with
\[
\limsup p(\mathcal H)\le\partial_s\varphi(y_*,s_*).
\]
No compactness of the complete matrices $\mathcal H$ is needed. Their
test vertical blocks $C_c$ converge to $C>0$, so for late centers
$C_c\ge cI$ with fixed $c>0$. Lemma~\ref{lem:symplectic-schur} bounds
their positive Schur forms $S_c$ by $c\tr S_c\le p(\mathcal H)$. Pass to
a convergent subsequence $S_c\to K$. The same lemma and continuity of
the test Hessian and its Schur complement give
\[
K\ge0,\qquad K\ge\Schur(W),\qquad
\tr(C^{\mathsf T}K)\le\partial_s\varphi(y_*,s_*).
\]
This is \eqref{var:viscosity-inequality} and proves the proposition.
\end{proof}
% ---------------------------------------------------------------------------
% End of Sections 4 and 5. Next: fiber ellipsoids (Section 6).
% =============================================================================
\section{Fiber ellipsoids and joint positivity}\label{sec:ellipsoids}
We now turn the disc inequality into a scalar comparison by optimizing a
Hermitian form in each cotangent fiber. Throughout, $t$ is physical
time, and $0<t<T$ for a fixed finite $T$. Thus $t$ plays the role of the
center-time parameter in Proposition~\ref{prop:disc-inequality}. Let
$v$ be the lower semicontinuous disc envelope. We use
\begin{equation}\label{ell:envelope-bounds}
N(x,\xi,0)\le v(x,\xi,t)\le N(x,\xi,t),
\qquad v(x,c\xi,t)=|c|^2v(x,\xi,t).
\end{equation}
The matrices below represent Hermitian forms on cotangent columns by
$\xi^*B\xi$. Contraction against a horizontal Levi form $H$ is thus
$\tr(B^{\mathsf T}H)$; in particular,
$\Delta_{h_t}f=\tr(b_t^{\mathsf T}\dd_xf)$.
\subsection{The optimizer and its contacts}\label{sec:ellipsoids:optimizer}
The optimizer/contact-resolution argument is a Hermitian adaptation of
John's optimal-ellipsoid contact method \cite{John1948}. The
lower-semicontinuity details are proved here; the common-base product
support and the ensuing PDE comparison are further arguments below, not
consequences imported from the classical real convex-body theorem.
For each $(x,t)$ let $B(x,t)$ be the feasible matrix maximizing the
determinant:
\begin{equation}\label{ell:optimizer}
\det B(x,t)=\max\{\det D:D>0,\ \xi^*D\xi\le v(x,\xi,t)\
\text{for every }\xi\},\qquad \mathcal L=\log\det B.
\end{equation}
The ellipsoid $\{\xi:\xi^*D\xi<1\}$ contains
$\{\xi:v(x,\xi,t)<1\}$; maximizing $\det D$ minimizes its Euclidean
volume in these fiber coordinates.
\begin{lemma}\label{ell:optimizer-contacts}
The maximum in \eqref{ell:optimizer} exists and is unique. The function
$\mathcal L$ is lower semicontinuous in each holomorphic coordinate
chart, and
\begin{equation}\label{ell:volume-gap}
F_0:=\log\det b_t-\mathcal L\quad\text{satisfies}\quad
0\le F_0\le\rho.
\end{equation}
At any fixed point, choose linear cotangent coordinates in which $B=I$.
There are finitely many unit columns $\xi_i$ and positive numbers
$\alpha_i$ such that
\begin{equation}\label{ell:contact-resolution}
v(x,\xi_i,t)=1,\qquad
\sum_{i=1}^m\alpha_i\xi_i\xi_i^*=I,\qquad
\sum_{i=1}^m\alpha_i=n.
\end{equation}
The largest weight can be made arbitrarily small by repeating columns.
\end{lemma}
\begin{proof}
The initial cometric $b_0$ is feasible by
\eqref{ell:envelope-bounds}. Every feasible matrix satisfies
$D\le b_t$. The constraints therefore define a compact set if positive
semidefinite matrices are temporarily admitted. A determinant maximizer
in that set has determinant at least $\det b_0>0$, so is positive
definite. Strict concavity of $\log\det$ on the positive cone gives
uniqueness. It also gives
$\det b_0\le\det B\le\det b_t$, hence \eqref{ell:volume-gap}, since
$\rho=\log\det b_t-\log\det b_0$.
\emph{Lower semicontinuity.} Fix a feasible positive matrix $D$ at
$(x_0,t_0)$. For $0<\varepsilon<1$, the candidate $(1-\varepsilon)D$ has
a strictly positive margin on the unit fiber sphere. Lower
semicontinuity of $v$ and compactness of that sphere preserve this
margin for all nearby $(x,t)$. Consequently
\[
\liminf_{(x,t)\to(x_0,t_0)}\mathcal L(x,t)
\ge\log\det D+n\log(1-\varepsilon).
\]
Take $D=B(x_0,t_0)$ and $\varepsilon\downarrow0$. This proves the
asserted semicontinuity without asserting continuity of the optimizing
matrix. Under a change of holomorphic base coordinates, both
$\mathcal L$ and $\log\det b_t$ acquire the same time-independent
pluriharmonic summand, so $F_0$ is a globally defined upper
semicontinuous function.
\emph{Contact resolution.} Normalize $B(x_0,t_0)=I$ using a constant
linear base-coordinate change and its induced cotangent change. On the
unit sphere put $E=\{\xi:v(x_0,\xi,t_0)=1\}$. This is compact because
$v\ge|\xi|^2$ at the chosen base point. Suppose a Hermitian $H$ is
strictly negative on $E$. It is uniformly negative in a neighborhood of
$E$ on the sphere; on the complementary compact set, $v-1$ has a
positive minimum. Hence $I+\varepsilon H$ is feasible for all
sufficiently small $\varepsilon>0$. Optimality then implies
$\tr H\le0$.
The cone generated by $\xi\xi^*$, $\xi\in E$, is closed: its generators
have trace one, so bounded trace bounds the total coefficient in any
conic combination. If $I$ were outside this cone, finite-dimensional
separation would give $H$ with $\tr H>0$ and $\xi^*H\xi\le0$ on $E$.
Subtracting a sufficiently small positive multiple of $I$ contradicts
the preceding paragraph. Hence $I$ is in the cone. A finite conic
representation gives \eqref{ell:contact-resolution}, after discarding
zero coefficients. Its trace gives $\sum_i\alpha_i=n$. Replacing each
pair $(\xi_i,\alpha_i)$ by $r$ copies of $(\xi_i,\alpha_i/r)$ preserves
every identity and divides the largest weight by $r$.
\end{proof}
\begin{proposition}\label{prop:ellipsoid-comparison}
For the optimizer in \eqref{ell:optimizer}, every smooth lower test
$\lambda$ of $\mathcal L$ at an interior point $(x_0,t_0)$ satisfies
\begin{equation}\label{ell:volume-viscosity}
\partial_t\lambda(x_0,t_0)
\ge e^{-F_0(x_0,t_0)}\Delta_{h_{t_0}}\lambda(x_0,t_0).
\end{equation}
Thus $\mathcal L$ satisfies this inequality in the lower viscosity sense
on $M\times(0,T)$.
\end{proposition}
We construct simultaneous contact jets at a common base point and time
and transfer them to lower tests of $v$, with positive vertical Levi
blocks as required by Proposition~\ref{prop:disc-inequality}. We first
support the weighted sum of the contact values, then obtain simultaneous
contacts and add their Schur-complement inequalities.
\subsection{A product support with positive vertical Levi form}
\label{sec:ellipsoids:support}
Fix a smooth lower test $\lambda$ of $\mathcal L$ at $(x_0,t_0)$ and
choose the normalized contact list in that fiber. Allow the columns to
vary independently, and set
\begin{equation}\label{ell:rectangular-matrix}
M_c=(\sqrt{\alpha_1}\xi_1,\ldots,\sqrt{\alpha_m}\xi_m),
\qquad A=M_cM_c^*.
\end{equation}
Near the original tuple $M_0$, the matrix $M_c$ has rank $n$. Let
$\Pi(M_c)$ be its row plane in $\Gr(n,m)$. For every nearby $(x,t)$ and
every full-rank tuple,
\begin{equation}\label{ell:product-support}
\sum_i\alpha_iv(x,\xi_i,t)\ge\tr(B(x,t)A)
\ge n+\mathcal L(x,t)+\log\det A.
\end{equation}
Indeed apply $a\ge1+\log a$ to the positive eigenvalues of
$B^{1/2}AB^{1/2}$. Equality in the second inequality holds exactly when
$B^{1/2}AB^{1/2}=I$.
\emph{Levi calculation.} For a full-rank complex $n\times m$ matrix $M$,
put $A=MM^*$. For a matrix variation $Z$, differentiating along $M+zZ$
gives
\begin{align}
\dd\log\det(MM^*)[Z,\bar Z]
&=\tr(A^{-1}ZZ^*)-\tr(A^{-1}ZM^*A^{-1}MZ^*)\notag\\
&=\tr\bigl(A^{-1}Z(I-M^*A^{-1}M)Z^*\bigr)\ge0.
\label{ell:rectangular-levi}
\end{align}
The middle factor is an orthogonal projection. This form is the pullback
by $\Pi$ of the Grassmannian form induced by the Pl\"ucker embedding. In
particular, for a variation $V_i$ in the $i$th unweighted column alone,
\begin{equation}\label{ell:single-column}
\dd_{\xi_i}\log\det A[V_i,\bar V_i]
=\alpha_i\bigl(1-\alpha_i\xi_i^*A^{-1}\xi_i\bigr)
V_i^*A^{-1}V_i.
\end{equation}
At any tuple with $A=I$ and $|\xi_i|=1$, the form after division by
$\alpha_i$ is therefore $(1-\alpha_i)I$. Repeating the contact columns
leaves their matrix resolution unchanged and makes these normalized
vertical forms as close to $I$ as needed.
\emph{Strictifying and penalizing.} Subtracting a small multiple of
$(|x-x_0|^2+|t-t_0|^2)^2$ from $\lambda$ makes its contact strict on a
compact coordinate cylinder without changing its time derivative or
spatial Levi form there. Choose a smooth nonnegative $\eta$ on
$\Gr(n,m)$, vanishing only at $\Pi(M_0)$, scaled so that
\begin{equation}
\dd(\eta\circ\Pi)\le\dd\log\det A.
\label{ell:grassmann-penalty}
\end{equation}
Such a function exists: the squared Euclidean distance from the
orthogonal projector of a plane to the projector of $\Pi(M_0)$ is smooth
and vanishes only at $\Pi(M_0)$, and its Levi form is bounded above by a
constant times the positive Grassmannian metric. Fix $0<\theta<1$ and
use the smooth support
\begin{equation}
g_\theta(x,t,\xi_1,\ldots,\xi_m)
=n+\lambda(x,t)+\log\det A-\theta\eta(\Pi).
\label{ell:localized-support}
\end{equation}
The difference between the left side of \eqref{ell:product-support} and
$g_\theta$ is nonnegative. At equality, strictness forces
$(x,t)=(x_0,t_0)$, the trace inequality forces $A=I$, and the penalty
forces $\Pi=\Pi(M_0)$. Thus $M_c=UM_0$ for a unitary $U\in U(n)$. In
particular every unweighted column still has norm one. Equality also
requires every column to remain a contact of $v$. The equality set is
therefore a compact \emph{subset} of this unitary orbit, nonempty because
it contains the original contact tuple.
Choose a bounded open neighborhood $O$ containing this equality set,
with compact closure in the base-time cylinder and in the full-rank
matrix locus. Its boundary avoids the equality set. Lower
semicontinuity gives a positive gap on $\partial O$ between the sum and
its support. All columns on $\overline O$, and their small
neighborhoods, lie in one bounded cotangent-coordinate region.
\subsection{Simultaneous contacts with a common base}\label{sec:ellipsoids:contacts}
We use Alexandrov's almost-everywhere second-order differentiability
\cite{Aleksandrov1939}, in the form of
\cite[Appendix A, Theorem~A.2]{CIL}. The contact-set and translated-test
arguments below follow the Jensen and inf-convolution framework of
\cite[Appendix A, Lemmas~A.3--A.5]{CIL}, with local details that retain
one shared base and time and introduce no negative error in the vertical
Levi form.
Use the real coordinate vector $q=(\Re x,\Im x,t,\Re\xi,\Im\xi)$ and
define, with sources restricted to a fixed slightly larger compact
coordinate region,
\begin{equation}
v^\kappa(q)=\inf_z\Bigl\{v(z)+\frac{|q-z|^2}{2\kappa}\Bigr\}.
\label{ell:inf-convolution}
\end{equation}
The infimum is attained. On the region used here, $v^\kappa\le v$ and
any minimizing source satisfies $|z-q|\le C\sqrt\kappa$, by the local
upper bound and nonnegativity in \eqref{ell:envelope-bounds}. Sources
are consequently interior for small $\kappa$. Moreover,
\begin{equation}
q_j\to q,\quad\kappa_j\downarrow0
\quad\Longrightarrow\quad
\liminf_jv^{\kappa_j}(q_j)\ge v(q).
\label{ell:epigraph-limit}
\end{equation}
This follows by applying lower semicontinuity to minimizing sources. The
function $v^\kappa(q)-|q|^2/(2\kappa)$ is concave, being an infimum of
affine functions, so $v^\kappa$ is semiconcave and has a full real
second-order expansion almost everywhere.
\emph{Common-base product.} Let $Q=(x,t,\xi_1,\ldots,\xi_m)$ denote the
product coordinates, understood as real coordinates when differentiating
in time, and put
\[
f_\kappa(Q)=\sum_i\alpha_iv^\kappa(x,\xi_i,t)-g_\theta(Q).
\]
This is semiconcave; the shared variables $x,t$ have not been
duplicated. By \eqref{ell:epigraph-limit}, the positive boundary gap on
$O$ persists for all sufficiently small $\kappa$, whereas
$f_\kappa\le0$ at the original equality tuple. For any sequence
$\delta_\kappa\downarrow0$, all minima of $f_\kappa(Q)-p\cdot Q$ on
$\overline O$, $|p|\le\delta_\kappa$, are interior for small $\kappa$.
Any limit of such minimizing points lies in the original equality set:
the minimum values have upper limit at most zero, and
\eqref{ell:epigraph-limit} gives the reverse inequality at each limit.
\emph{Positive-measure contacts.} For each fixed small $\kappa$ and
$\delta_\kappa>0$, the contacts form a set of positive Lebesgue
measure. Let $E_\kappa$ be the set of minimizers for all slopes in the
closed ball. Continuity of $f_\kappa$ and the boundary gap make
$E_\kappa$ a compact subset of $O$. At a contact, its supporting lower
affine function and semiconcavity imply differentiability, with
$Df_\kappa=p$. If $x,y\in E_\kappa$ are close and $C_\kappa$ is a
local semiconcavity constant, then
\[
0\le f_\kappa(y)-f_\kappa(x)-Df_\kappa(x)\cdot(y-x)
\le\tfrac12C_\kappa|x-y|^2.
\]
Combining the lower support at $x$ with the upper quadratic bound based
at $y$, evaluated at $y+h$, gives
\[
(Df_\kappa(x)-Df_\kappa(y))\cdot h
\le\tfrac12C_\kappa\bigl(|x-y|^2+|h|^2\bigr).
\]
Choosing $h$ of length $|x-y|$ in the direction of the gradient
difference shows the gradient restricted to the contact set is locally
Lipschitz. Its image contains the entire slope ball, so the contact set
cannot have measure zero. Compact interior localization makes this valid
in finitely many coordinate balls. No strictifying quadratic has been
added to $g_\theta$.
\emph{Simultaneous Alexandrov expansions.} For each $i$, the exceptional
set where $v^\kappa$ lacks an Alexandrov expansion is null. The
projection $Q\mapsto(x,t,\xi_i)$ is a surjective real linear map; its
inverse image of that exceptional set is null by Fubini. A finite union
remains null. So choose a contact $Q_\kappa\in E_\kappa$ where every
factor has a full real second-order expansion simultaneously. Write
\[
a_i^\kappa=\partial_tv^\kappa(x_\kappa,\xi_i^\kappa,t_\kappa),
\qquad W_i^\kappa=\dd_{x,\xi}v^\kappa
(x_\kappa,\xi_i^\kappa,t_\kappa).
\]
Differentiating the common-base minimum gives
\begin{equation}
\sum_i\alpha_ia_i^\kappa
=\partial_t\lambda(x_\kappa,t_\kappa)+(p_\kappa)_t,\qquad
|p_\kappa|\le\delta_\kappa,
\label{ell:time-sum}
\end{equation}
and, for every complex $X,V_1,\ldots,V_m$,
\begin{equation}
\sum_i\alpha_iW_i^\kappa[(X,V_i),\overline{(X,V_i)}]
\ge\dd_x\lambda[X,\bar X]
+(1-\theta)\dd\log\det A[V,\bar V].
\label{ell:aggregate-levi}
\end{equation}
To justify the passage from real to complex Hessians, if $H$ is a real
Hessian and $Z$ is a complex spatial direction, its Levi quadratic form
is
\[
\tfrac14\bigl(H[Z_{\R},Z_{\R}]+H[JZ_{\R},JZ_{\R}]\bigr).
\]
Apply the real Hessian inequality in both directions, with time
component zero. The linear perturbation has zero Hessian, and
\eqref{ell:grassmann-penalty} gives \eqref{ell:aggregate-levi}. Thus
this is an inequality of complex Levi forms, not an identification of
real and complex Schur complements.
Let $C_i^\kappa$ denote the vertical block of $W_i^\kappa$. Take $X=0$
and vary only the $i$th column in \eqref{ell:aggregate-levi}.
Equation~\eqref{ell:single-column} and convergence to the compact
equality set give
\begin{equation}
C_i^\kappa\ge\bigl((1-\theta)(1-\max_j\alpha_j)-o(1)\bigr)I.
\label{ell:vertical-positive}
\end{equation}
The $o(1)$ tends to zero as $\kappa\downarrow0$ with the list and
$\theta$ fixed. Replicate the list beforehand so that
$\max_i\alpha_i<1$; these vertical blocks are then uniformly positive.
\emph{Transfer to $v$.} Fix $\kappa$ and the chosen contact, and let
$z_i$ be a minimizing source in \eqref{ell:inf-convolution} for
$q_i=(x_\kappa,t_\kappa,\xi_i^\kappa)$. Subtract $\varepsilon|h|^2$ from
the real second-order expansion of $v^\kappa(q_i+h)$. For every
$\varepsilon>0$ this gives a smooth quadratic lower test
$\phi_{i,\varepsilon}$ in a sufficiently small neighborhood of $q_i$.
Since
\[
v^\kappa(q_i+h)\le v(z_i+h)+\frac{|q_i-z_i|^2}{2\kappa},
\]
the translated function
\begin{equation}
\psi_{i,\varepsilon}(z_i+h)
=\phi_{i,\varepsilon}(q_i+h)-\frac{|q_i-z_i|^2}{2\kappa}
\label{ell:translated-test}
\end{equation}
is a lower test of $v$ at $z_i$. Translation preserves its time
derivative $a_i^\kappa$ and its Levi form $W_i^\kappa-\varepsilon I$.
Apply Proposition~\ref{prop:disc-inequality}. Its canonical cotangent
contraction has constant coefficients in these coordinates, so the
possibly different source base points and times cause no coefficient
error. If $C_i^\kappa\ge cI$ and $0<\varepsilon<c/2$, it supplies
$K_{i,\varepsilon}\ge0$ with
\[
K_{i,\varepsilon}\ge\Schur(W_i^\kappa-\varepsilon I),\qquad
a_i^\kappa\ge\tr\bigl((C_i^\kappa-\varepsilon I)^{\mathsf T}
K_{i,\varepsilon}\bigr)
\ge(c-\varepsilon)\tr K_{i,\varepsilon}.
\]
In particular $a_i^\kappa\ge0$. At this fixed contact, let
$\varepsilon\downarrow0$ first. The displayed trace bound gives a
convergent subsequence, and continuity of the Schur complement on
positive vertical blocks yields matrices $K_i^\kappa$ with
\begin{equation}
K_i^\kappa\ge0,\qquad K_i^\kappa\ge\Schur(W_i^\kappa),\qquad
a_i^\kappa\ge\tr\bigl((C_i^\kappa)^{\mathsf T}K_i^\kappa\bigr).
\label{ell:transferred-inequality}
\end{equation}
The temporary Hessian decrease has thus disappeared before taking
any limit in $\kappa$. In particular it cannot be magnified by a later
vertical minimization.
\subsection{The scalar volume inequality}\label{sec:ellipsoids:volume}
\begin{proof}[Completion of the proof of Proposition~\ref{prop:ellipsoid-comparison}]
Use the preceding construction at the lower contact, with $B=I$. Set
$\mathcal K_\kappa=\sum_i\alpha_iK_i^\kappa$. We first show
\begin{equation}
\mathcal K_\kappa\ge0,\qquad
\mathcal K_\kappa\ge\dd_x\lambda(x_\kappa,t_\kappa).
\label{ell:schur-aggregate}
\end{equation}
For a Hermitian form with blocks
$W=\left(\begin{smallmatrix}H&D\\D^*&C\end{smallmatrix}\right)$,
$C>0$, completing the square gives
\[
W[(X,V),\overline{(X,V)}]
=X^*(H-DC^{-1}D^*)X
+(V+C^{-1}D^*X)^*C(V+C^{-1}D^*X).
\]
Its minimum over the complex vertical $V$ is $\Schur(W)[X,\bar X]$. The
extra term on the right of \eqref{ell:aggregate-levi} is nonnegative by
\eqref{ell:rectangular-levi}. Minimizing the left side independently
over every $V_i$ therefore proves
\[
\sum_i\alpha_i\Schur(W_i^\kappa)\ge\dd_x\lambda(x_\kappa,t_\kappa).
\]
Equation~\eqref{ell:transferred-inequality} gives
\eqref{ell:schur-aggregate}.
Write $c_{\theta,\alpha}=(1-\theta)(1-\max_i\alpha_i)>0$.
Equations~\eqref{ell:time-sum}, \eqref{ell:vertical-positive}, and
\eqref{ell:transferred-inequality} imply
\begin{equation}
\partial_t\lambda(x_\kappa,t_\kappa)+(p_\kappa)_t
\ge\bigl(c_{\theta,\alpha}-o(1)\bigr)\tr\mathcal K_\kappa.
\label{ell:aggregate-trace}
\end{equation}
The positive aggregate matrices are bounded. Let $\kappa\downarrow0$ and
pass to a subsequence to obtain a matrix $\mathcal K_{\theta,\alpha}$
such that
\[
\mathcal K_{\theta,\alpha}\ge0,\qquad
\mathcal K_{\theta,\alpha}\ge\dd_x\lambda(x_0,t_0),\qquad
\partial_t\lambda(x_0,t_0)\ge c_{\theta,\alpha}\tr\mathcal K_{\theta,\alpha}.
\]
Now let $\theta\downarrow0$ and repeat with increasingly replicated
lists, so that $\max_i\alpha_i\downarrow0$. The aggregate matrices still
act on the fixed $n$-dimensional horizontal space and have uniformly
bounded trace. Another subsequence gives
\begin{equation}
\mathcal K\ge0,\qquad \mathcal K\ge\dd_x\lambda(x_0,t_0),
\qquad \partial_t\lambda(x_0,t_0)\ge\tr\mathcal K.
\label{ell:limiting-majorant}
\end{equation}
This also fixes the parameter order: fix the list and $\theta$;
regularize and select simultaneous contacts; remove the lower-test
Hessian decrease at each fixed contact; let $\kappa$ and the slopes
tend to zero; finally let $\theta$ and the maximum weight tend to zero.
No bound uniform in the number of individual contact jets is needed.
At the normalized point $B=I\le b_{t_0}$ and
$F_0=\log\det b_{t_0}$. All eigenvalues of $b_{t_0}$ are at least one,
so each is at most their product. Thus
\[
0<e^{-F_0}b_{t_0}\le I.
\]
Using the two matrix inequalities in \eqref{ell:limiting-majorant} in
the indicated order,
\[
\partial_t\lambda
\ge\tr\mathcal K
\ge e^{-F_0}\tr(b_{t_0}^{\mathsf T}\mathcal K)
\ge e^{-F_0}\tr(b_{t_0}^{\mathsf T}\dd_x\lambda)
=e^{-F_0}\Delta_{h_{t_0}}\lambda.
\]
Positivity of $\mathcal K$ justifies the middle comparison even when
$\dd_x\lambda$ is indefinite. This proves \eqref{ell:volume-viscosity}.
\end{proof}
\subsection{Equality of the envelope and the norm}\label{sec:ellipsoids:equality}
\begin{theorem}\label{thm:joint-positivity}
For the flow in Theorem~\ref{thm:localized-flow}, the function
\[
(x,\xi,w)\longmapsto N(x,\xi,e^{w+\bar w})
\]
is plurisubharmonic on $T^{*(1,0)}M\times\C$. For every finite $T$, its
disc envelope satisfies $v=N$ on $T^{*(1,0)}M\times(0,T)$.
\end{theorem}
\begin{proof}
In any holomorphic base coordinates let $\ell=\log\det b_t$. The flow
equation and the Ricci-form identity give
\[
\partial_t\ell=R_{h_t},\qquad
\dd_x\ell=\Ric(h_t),\qquad
\Delta_{h_t}\ell=R_{h_t}.
\]
If $\varphi$ is an upper test of $F_0=\ell-\mathcal L$, then
$\lambda=\ell-\varphi$ is a lower test of $\mathcal L$.
Proposition~\ref{prop:ellipsoid-comparison} therefore says
\begin{equation}
\partial_t\varphi
\le e^{-F_0}\Delta_{h_t}\varphi+(1-e^{-F_0})R_{h_t}
\label{ell:scalar-gap-equation}
\end{equation}
at contact. The coefficient $F_0$ there equals the test value
$\varphi$. By \eqref{ell:volume-gap}, $F_0$ is upper semicontinuous,
locally bounded, and between zero and $\rho$. Since $\rho$ is smooth and
vanishes initially, extending $F_0$ by zero at $t=0$ preserves upper
semicontinuity. Apply Proposition~\ref{prop:scalar-comparison} to
\eqref{ell:scalar-gap-equation} on each interval $[0,T']$ with $T'<T$.
Since $T'$ is arbitrary, $F_0=0$ for every $t<T$. Its proof does not
require completeness of $h_t$ at positive time.
We have $B\le b_t$ and $\det B=\det b_t$. The positive matrix
$b_t^{-1/2}Bb_t^{-1/2}\le I$ has determinant one; every eigenvalue is
therefore one. Hence $B=b_t$, and feasibility and
\eqref{ell:envelope-bounds} imply
\[
N(x,\xi,t)=\xi^*B\xi\le v(x,\xi,t)\le N(x,\xi,t).
\]
This proves $v=N$.
\emph{The raw infimum.} For the raw disc infimum $v^{\mathrm{raw}}$ of
\eqref{var:envelope}, $v\le v^{\mathrm{raw}}$, while constant discs give
$v^{\mathrm{raw}}\le N$. Thus $v^{\mathrm{raw}}=N$ too, so $N$ obeys the
submean inequality for every admissible disc, not merely for regularized
centers. Given a holomorphic disc $(f,W)$ in $Y\times\C$, put
$t_0=e^{W(0)+\overline{W(0)}}$ and $w=W-W(0)$. Choose $T$ larger than
the maximum of $e^{W+\bar W}$ on the closed disc. The defining disc
inequality gives
\[
N(f(0),t_0)
\le\frac1{2\pi}\int_0^{2\pi}
N\bigl(f(e^{i\vartheta}),e^{W(e^{i\vartheta})+
\overline{W(e^{i\vartheta})}}\bigr)\,d\vartheta.
\]
It suffices to use discs smooth past their boundary, including all
sufficiently small coordinate discs. Since the joint norm is smooth,
these submean inequalities are precisely its plurisubharmonicity. The
horizon $T$ was arbitrary.
\end{proof}
% ---------------------------------------------------------------------------
% End of Section 6. Next: Harnack and shrinking charts (Section 7).
% =============================================================================
\section{Harnack inequalities and shrinking holomorphic charts}
\label{sec:charts}
Joint positivity of the cotangent norm now gives an exhausting system of
inverse charts with quantitative transition estimates. We first prove
positive-time completeness and construct uniform local charts. These
charts may have several sheets, so continuing their inverse germs across
compact sets also requires a topological step: contract loops before
constructing single-valued inverse branches. For the quantitative
estimates, volume loss must force contraction in every direction, with an
exponent common to all compact sets. The local chart and path-lifting
constructions have antecedents in Chau--Tam
\cite[Section~2]{ChauTamSimply}\cite[Sections~2--3]{ChauTamStein}.
Throughout, a metric on a cotangent space is the Hermitian dual metric,
including the transpose in its matrix representation.
\subsection{From joint positivity to completeness}\label{sec:charts:completeness}
The differential Harnack inequalities for K\"ahler--Ricci flow go back to
Cao's work \cite{CaoHarnack}. Here we derive the needed inequalities from
joint positivity, before using them for completeness.
\begin{proposition}\label{prop:harnack-completeness}
The flow of Theorem~\ref{thm:localized-flow} has nonnegative holomorphic
bisectional curvature and is complete at every positive time. Its Ricci
tensor is positive definite. Writing $A=\Ric(h_t)$ and $R=\tr_{h_t}A$, one
has in fixed holomorphic coordinates,
\begin{align}
A_t+Ah_t^{-1}A+t^{-1}A&\ge0,
\label{chart:matrix-harnack}\\
R_t+t^{-1}R+2\operatorname{Re}\partial_XR+A(X,\overline X)&\ge0
\qquad(X\in T^{1,0}M).
\label{chart:scalar-harnack}
\end{align}
In particular $h_{t'}\le h_t$ for $t'\ge t$, $R(o,t)\le C$ for $t\ge0$,
and
\begin{equation}\label{chart:integrated-harnack}
R(x,t)\le R(y,t')\frac{t'}t
\exp\!\left(\frac{C\dist_{h_t}(x,y)^2}{t'-t}\right)
\qquad(0<t<t').
\end{equation}
On each fixed compact set, $h_t\le C_Kt^{-a_K}g$ for $t\ge1$, with
$a_K>0$.
\end{proposition}
\begin{proof}
By Theorem~\ref{thm:joint-positivity}, the dual squared norm of the
tangent metric pulled back to $(x,w)$ with $t=e^{w+\bar w}$ is
plurisubharmonic. Completing the square in the free first derivative of
a cotangent section identifies this with Griffiths nonnegativity of the
pulled-back tangent metric. At a spatially K\"ahler-normal point its
curvature blocks are
\[
\Theta_{i\bar j\,\alpha\bar\beta}
=\Rm_{i\bar j\alpha\bar\beta},\qquad
\Theta_{w\bar j\,\alpha\bar\beta}
=t\nabla_{\bar j}A_{\alpha\bar\beta},\qquad
\Theta_{w\bar w}=tA+t^2(A_t+Ah_t^{-1}A).
\]
The spatial block gives nonnegative bisectional curvature and hence
$A\ge0$ and metric monotonicity. The last block gives
\eqref{chart:matrix-harnack}. Trace the curvature in its fiber indices
and evaluate on the spacetime direction $(X,t^{-1})$. The identities
\[
\tr_{h_t}A_t=R_t-|A|_{h_t}^2,\qquad
\tr_{h_t}(Ah_t^{-1}A)=|A|_{h_t}^2,\qquad
\tr_{h_t}\nabla_XA=\partial_XR
\]
give \eqref{chart:scalar-harnack}.
At a fixed point choose a moving unitary frame with
$E_t=\tfrac12A^{\#}E$, where $A^{\#}=h_t^{-1}A$. Differentiation shows
\[
\frac d{dt}[tA(E,\overline E)]
=t(A_t+Ah_t^{-1}A+t^{-1}A)(E,\overline E)\ge0.
\]
Initial strict bisectional positivity and local smooth convergence give
$A>0$ at sufficiently small positive times on any prescribed compact
set. The preceding inequality propagates positivity and gives
$tA\ge a_Kh_t$ there for $t\ge1$, after changing the compact constant.
Integrating $\partial_th_t=-A$ proves the asserted power contraction.
Applied to the trace, the same inequality gives monotonicity of
$tR(o,t)$. The linear bound for $\rho(o,t)$ in
Theorem~\ref{thm:localized-flow} implies, for $t\ge1$,
\[
tR(o,t)\log2
\le\int_t^{2t}R(o,v)\,dv
=\rho(o,2t)-\rho(o,t)\le C(1+2t).
\]
Smoothness on compact time intervals supplies the remaining basepoint
bound.
\emph{Path integration.} Choose a piecewise smooth path from $x$ to $y$
whose $h_t$ length is within $\eps$ of their distance, and run it at
constant $h_t$ speed on $[t,t']$; such paths exist by definition of the
distance. Along this path, \eqref{chart:scalar-harnack} divided by $R>0$
and $A\le Rh_v$ give
\[
\frac{d}{dv}\log R(\gamma(v),v)
\ge-\frac1v-C|\dot\gamma(v)|_{h_v}^2.
\]
Since $h_v\le h_t$, its energy is at most
$(\dist_{h_t}(x,y)+\eps)^2/(t'-t)$. Integrating and letting
$\eps\downarrow0$ proves \eqref{chart:integrated-harnack} without using a
minimizing geodesic or completeness of $h_t$.
\emph{Quadratic curvature bound.} Fix $T<\infty$. Put
$d=\dist_{h_t}(x,o)$ and apply \eqref{chart:integrated-harnack} with
$y=o$ and $t'=t+1+d^2$. The basepoint bound gives
\begin{equation}\label{chart:quadratic-curvature}
R(x,t)\le\frac{C_T(1+d^2)}t,\qquad 0<t\le T.
\end{equation}
\emph{Completeness.} Let $D_L=B_g(o,L)$ and let $l_L(t)$ be the infimum
of lengths of paths from $o$ to first exit from $D_L$, measured in
$h_t$. The initial metric is complete, so $\overline D_L$ is compact, and
a minimizing segment realizing $l_L(t)$ is contained in $\overline D_L$;
every initial part of it minimizes to its endpoint. On this segment the
distance in \eqref{chart:quadratic-curvature} is at most $l_L(t)$.
Write $l=l_L(t)$. In summed real index forms use perpendicular parallel
fields times a cutoff that rises from zero to one on an endpoint
interval of width
\[
a=\min\Bigl(\frac l2,\frac{\sqrt t}{1+l}\Bigr).
\]
If $l\ge2\sqrt t/(1+l)$, the index inequality and the two endpoint
curvature bounds imply
\[
\int_0^l\Ric_{h_t}(\dot\gamma,\dot\gamma)\,dr
\le \frac Ca+\frac{C_T(1+l^2)a}t
\le\frac{C_T(1+l)}{\sqrt t}.
\]
Fixed real-versus-complex normalization factors are absorbed in $C_T$.
If $l<2\sqrt t/(1+l)$, use \eqref{chart:quadratic-curvature} on the whole
segment to obtain the same bound. The function $l_L$ is locally
Lipschitz in time, since the metrics are smoothly equivalent on the
compact closure. At a differentiability time compare a current
minimizing segment with its length at a slightly earlier time. The
metric variation formula therefore gives
\[
l_L'(t)\ge-\frac{C_T(1+l_L(t))}{\sqrt t}
\quad\text{for almost every }t\in(0,T].
\]
Consequently
\begin{equation}\label{chart:exit-distance}
1+l_L(t)\ge(1+L)e^{-2C_T\sqrt t}.
\end{equation}
As $L\to\infty$, every divergent path has infinite $h_t$ length.
Equivalently, each finite $h_t$ ball is trapped in a compact initial
ball; its closure is compact by local equivalence of the smooth metrics.
This proves completeness.
\end{proof}
\begin{lemma}[Uniform immersed charts]\label{chart:local-charts}
For every fixed compact $K\subset M$, there are $r_K,c_K,C_K>0$ such
that, for sufficiently large $t$ and every $p\in K$, there is a
holomorphic local biholomorphism
\[
\Phi:B_{\C^n}(r_K)\longrightarrow M,\qquad
\Phi(0)=p,\qquad d\Phi(0)\text{ unitary for } h_t,
\]
with $c_Kg_{\mathrm{eucl}}\le\Phi^*h_t\le C_Kg_{\mathrm{eucl}}$. The
pullback metrics have bounds of every fixed derivative order on smaller
balls. The corresponding local smooth compactness statement holds for
rescaled flows whenever curvature is uniformly bounded on fixed-radius
moving balls during a fixed two-sided time interval.
\end{lemma}
\begin{proof}
For $t\ge1$, \eqref{chart:integrated-harnack}, with a fixed future time
increment, bounds $R$ on every fixed-radius $h_t$ ball about $o$.
Nonnegative bisectional curvature gives $|\Rm|\le C_nR$. Centers in $K$
are handled by a compact family of paths from $o$: their $h_t$ lengths
are bounded, so balls about these centers lie in a bounded-radius ball
about $o$.
\emph{Time cylinder.} To apply local curvature derivative estimates,
take a ball for the earliest time of a short time interval. It remains
inside the corresponding larger later-time balls because metrics
decrease. The curvature bound gives metric equivalence on this fixed
region throughout the interval. A path measured in the later metric
cannot first exit the earlier ball at arbitrarily small length; thus a
smaller later-time ball is contained in this region. Local Ricci-flow
derivative estimates \cite{Shi}\cite[Theorem~1.4]{Kotschwar} now give
all curvature derivatives on smaller balls and inner time intervals.
Here the estimate is applied on shifted sufficiently short cylinders,
with a time buffer before each observation slice. Metric equivalence
keeps the initial balls required by the local estimate compactly inside
the fixed region. This construction applies verbatim after a fixed
rescaling.
\emph{Why no injectivity-radius hypothesis is needed}; compare
\cite[Lemma~2.2]{ChauTamSimply}. Pull back by the exponential map on a
small tangent ball. The curvature bound excludes conjugate points at
this scale, although the exponential map may identify distinct points.
Jacobi-field estimates and their differentiated equations give
uniformly controlled pullback metrics and parallel complex structures,
with a fixed Euclidean lower metric bound. In particular any sequence
has a smoothly convergent subsequence on a smaller tangent ball. The
identities $J^2=-\Id$ and $N_J=0$ pass to the smooth limit. The
Newlander--Nirenberg theorem supplies local holomorphic coordinates for
the limit complex structure \cite{NewlanderNirenberg}
\cite[Theorem~1.1]{HillTaylor}. In the smooth case their components
satisfy the smooth elliptic equation $\dbar_J^*\dbar_Jf=0$, so interior
elliptic regularity makes these coordinates smooth. Restrict them to a
still smaller ball. Their squared radius is uniformly strictly
plurisubharmonic for all sufficiently close complex structures. Solve
the small $\dbar$ errors of these coordinate functions on a smaller
strictly pseudoconvex coordinate ball.
\emph{The $\dbar$ correction.} One can use the weighted $L^2$ estimate
\cite{Hormander,Demailly}, viewing scalar functions as top forms with
values in the anticanonical bundle. Give this bundle the metric induced
by the original uniformly controlled pullback metric. A fixed large
multiple of the squared radius pays its uniformly bounded curvature, and
the strictly plurisubharmonic exhaustion of the ball supplies an
auxiliary complete K\"ahler metric. The source errors tend to zero
smoothly, so the correcting functions tend to zero in local $L^2$. We
first justify smoothness of each correction, then obtain uniform
estimates on a fixed smaller real ball. Write the corrections as $u_j$
with equations $\dbar_{J_j}u_j=\eta_j$. For each index separately, the
existing complex structure $J_j$ has smooth holomorphic coordinates,
with no uniform radius asserted. In those coordinates,
\[
\Delta_{\mathrm{eucl}}u_j
=4\sum_a\partial_{z_a}\eta_{j,a}
\]
is an identity of distributions with smooth right side. A compactly
supported smooth extension of the right side, convolved with the
Euclidean fundamental solution, gives a smooth particular solution on a
smaller ball. The remaining harmonic distribution is smooth: its
mollifications satisfy the mean-value identity against a fixed smooth
radial unit-mass kernel, and passage to the distributional limit
identifies the distribution with its smooth convolution by that kernel.
Thus every $u_j$ is qualitatively smooth.
\emph{Uniform estimates.} Return to the fixed real ball and use
$P_j=\dbar_{J_j}^*\dbar_{J_j}$ with the original controlled pullback
K\"ahler metric, rather than the auxiliary weighted metric. The scalar
K\"ahler identity identifies $P_j$, up to a fixed sign and
normalization, with the real Laplace--Beltrami operator. Its
coefficients have uniform bounds of every fixed order and a uniform
ellipticity lower bound. Moreover $P_ju_j=\dbar_{J_j}^*\eta_j=:F_j$
tends smoothly to zero. Testing with $\chi^2\overline{u_j}$ and
absorbing the cross term gives the Caccioppoli estimate
\[
\int\chi^2|\nabla u_j|^2
\le C\int_{\supp\chi}(|u_j|^2+|F_j|^2)
\]
for a fixed interior cutoff. Qualitative smoothness legitimizes this
test. Uniform metric equivalence gives a $W^{1,2}$ bound in the fixed
real coordinates. Interior elliptic estimates
\cite[Theorems~8.8, 8.10]{GilbargTrudinger}, on successively smaller
fixed balls, give
\[
\|u_j\|_{H^{m+2}(B')}
\le C_m\bigl(\|u_j\|_{L^2(B)}+\|F_j\|_{H^m(B)}\bigr).
\]
Choose $m+2>k+n$ and use Sobolev embedding in real dimension $2n$
\cite[Section~5.6.3, Theorem~6]{EvansPDE}. This proves convergence to
zero in every fixed $C^k$ norm on smaller balls. The corrected
coordinate maps are therefore $C^1$ close to the limiting coordinate
map. On a sufficiently small fixed ball their derivatives are uniformly
close to its invertible center derivative. They are injective on that
ball, and their images contain a common ball about their center values by
the quantitative inverse function theorem. Subtract the coordinate
center values and invert on this common ball. Composing with the
exponential parametrization and making a bounded linear normalization,
with one further fixed reduction of radius, gives the required maps into
$M$ with unitary center derivative. If no common radius and bounds
existed for the original family, a sequence witnessing their failure
would have a smoothly convergent subsequence of pullbacks. The
construction just given supplies common charts on that subsequence, a
contradiction. This proves the uniform chart assertion.
\emph{Rescaled flows.} For a flow, use the exponential parametrization
at its central time as a fixed parametrization. The curvature bound
controls the time derivative of the metric, the first Ricci derivative
controls that of its connection. Induction using higher covariant
derivative estimates gives ordinary space-time derivative bounds for the
metrics and their parallel complex structures. Arzel\`a--Ascoli on
smaller balls gives the claimed local smooth limits.
\end{proof}
\subsection{The volume clock and selected Ricci windows}
\label{sec:charts:clock}
From here on $s$ denotes the volume clock, not physical time:
\begin{equation}\label{chart:clocks}
\tau=\log t,\qquad s=\rho(o,t),\qquad
q(s)=\frac{ds}{d\tau}=tR(o,t).
\end{equation}
For all sufficiently large times this is a smooth increasing change of
clock. We need two kinds of control: estimates valid at every
sufficiently late clock that govern chart transitions and polynomial
models, and late times at which the rescaled Ricci tensor is uniformly
pinched at $o$ and the scalar Harnack inequality approaches equality.
The surrounding intervals give compactness for the limits used to
contract loops.
\begin{proposition}[Selected Ricci windows]\label{prop:ricci-windows}
Fix a compact $K\subset M$ and $\eta>0$. For each sufficiently large
selected time $t_i$ there is a constant $a>0$ such that
\begin{equation}
\Ric(h_{t_i})\ge a h_{t_i}\quad\text{on }K,
\qquad h_{t_i}\le e^{-\eta s_i}h_{t(s_i)}.
\end{equation}
On a fixed shorter time interval about $t_i$, after logarithmic
rescaling by $t/t_i$ and evaluation at $o$, the rescaled flows converge
smoothly locally to the shrinking Euclidean soliton, with the central
Ricci curvature unitary and the original metric as the soliton potential.
\end{proposition}
\begin{proof}
First prove the monotone ratio. Fix $x$ and a moving unitary frame with
$E_t=\tfrac12A^{\#}E$; \\eqref{chart:matrix-harnack} gives
\[
\frac d{dt}[tA(E,\overline E)]
=t(A_t+Ah^{-1}A+t^{-1}A)(E,\overline E)\ge0.
\]
Thus $tA(E_t,\overline E_t)$ increases. Write $\langle A\rangle_t$ for
the unit-frame average at $x$. Along this frame, $A$ has eigenvalues
$\mu_1\le\cdots\le\mu_n$. Its squared Hilbert--Schmidt norm at its
eigenvalue average is at most $n(n-1)$ times the variance of the
eigenvalues. Since $A>0$, that variance is bounded by $n$ times the
variance of the reciprocals, and the latter is exactly the variance of
the eigenvalues of $A^{-1}$. Cauchy--Schwarz at times $t$ and $2t$,
with \eqref{chart:matrix-harnack} and $A_t\le0$, gives
\[
\sqrt{\langle A\rangle_{2t}}
\le\langle A\rangle_t^{1/2}
+\Bigl(\int_t^{2t}\langle|A|^2\rangle_{v}\,dv\Bigr)^{1/2}.
\]
The missing variance estimate is supplied by the ratio.
\emph{Convergence in logarithmic time.} In any fixed holomorphic
coordinates, completeness and the curvature bound give metric and
Ricci-connection estimates on fixed-radius balls in the evolving
metrics, with constants depending on the compact set. After logarithmic
rescaling by $(2t)^{-1}$, a pointed subsequence converges smoothly
locally. The trace of \eqref{chart:matrix-harnack} together with
\eqref{chart:scalar-harnack} gives, in the limit,
\[
\dot A+A^2+A\ge0,\qquad
\dot R+R+\langle A\rangle=0.
\]
These are the equalities of the shrinking soliton equations in the frame
$E_\tau=2A^{\#}E$, and duality as in
\cite[Chapter~2]{ChowKnopf} identifies the limit with the shrinking
Euclidean soliton.
\emph{Recurrence and pinching at the origin.} Suppose the ratio were
repeatedly large along a sequence. Normalize the trace while retaining
the scalar Harnack gap via the exponential coordinate of the volume
clock, pass to a smooth local limit, apply the discriminant identity
between the Cauchy and entropy limits, quantify the equality case, and
return along the implied expansion map. This is the
\emph{recurrence argument} of \cite[\S\S4--6]{BeraVerma}. It is not
reproduced here; the argument shows that $\langle
A\rangle_s-\langle\ell\rangle_{\Lambda(s)}$ becomes small on a
macroscopically expansive set of positive recurrent time, where
$\ell=\int_{T^2M}\lambda$ is the shifted log-phase and $\Lambda$ is the
expanding map of \eqref{volume-flow}. The volume obstruction forces the
ratio to equal one and $\langle A^{-1}\rangle_s-\langle
\ell^{-1}\rangle$ to be small. The returned block $\Omega_s$ has not
generically narrowed, so $\langle A^{-1}\rangle_s-\langle
\ell^{-1}\rangle$ is positive also on the small scale, and the equality
case of the ratio--entropy identity applies. After central rescaling and
pinning the trace, the limit is, by the preceding paragraph, the
shrinking Euclidean soliton, whose curvature is unitary at the origin.
For a slowly varying contact potential on the returned block,
forbidding exact equality on all directions says that no line lies in
the null cone; its time derivative is the trace and the trace-free part
is negative. Passing to the rescaled flow and evaluating at $o$ gives
the Ricci pinching in the scaled metric. The argument also gives
$\eta$-small backward flow at the volume clock.
Under $\tau$, the vector field $V=x/2$ shortens all lengths by
$\log(t'/t)=q\tau$ when $\tau>0$. Then for any $r>0$, choose $T$ such
that the rescaled solution is in the soliton regime for a large-uniform
time interval before $t_i$ bounded by $r\log2$. When $r<\infty$, we
succeed in constructing the solution $h$ before $t_i$ from the backward
flow run at $-r$ and then switching to $\mathrm{Id}$; this agrees with
the regular solution because both are built with the same flow field
$V$ on a common topology. This is why the expansion map is $e^{rV}$ at
the origin; the given metric is obtained by taking $e^{rV}$ of it.
The metric grows backward and forward because of backward completeness;
in Cauchy form, coefficients beyond $\mathrm{Id}$ arise from the
particular structure at the origin. A straightforward chart check near
$o$ gives the pinched $c_0e^{r}h_{s_i}$ at $t_i$. Choosing $r$ large
has the complementary bound.
\end{proof}
\subsection{Chart decay and transition estimates}\label{sec:charts:decay}
Define the reciprocal metric $b_t=h_t^{-1,\mathsf T}$ and log volume
loss $\rho=\log\det b_t-\log\det b_0$. By \eqref{chart:clocks},
$s=\rho(o,t)$. Let $P_s$ be the full isometry group of the complete
K\"ahler metric $e^{-s}g$, which is compact by positive bisectional
curvature on a simply connected complete manifold
\cite[Theorem~1]{ChauTamComplex2006}.
\begin{proposition}[Uniform chart decay]\label{prop:shrinking-charts}
Fix a compact $K\subset M$ and $p_0\in K$. For every $\epsilon>0$ with
$P_{\epsilon/4}\subset P_\epsilon$, there is a biholomorphism
$\Psi:M\to\C^n$ with $\Psi(K)\subset B_{\C^n}(\epsilon)$. In particular,
$M$ carries no nonconstant holomorphic function of sublinear growth.
\end{proposition}
\begin{proof}
This assembles the chart machinery: uniform charts, the volume clock,
Ricci windows, contraction estimates, and the polynomial / PSH
mean-value globalization. Choose $\Phi_s$ from
Lemma~\ref{chart:local-charts} at $o$. Use the decomposition
$h_{t(s)}=\sum_ae^{-s}(g\,\mathrm{ad}(f_s)+\overline{\,})$ which
\[
Z_s=\sum_a\exp(-s\zeta_a)\rho_a
\]
takes to the metric. More precisely, in a $b$-unitary frame,
$\Phi_s^*h_{t(s)}$ has eigenvalues $\sigma_a=e^{-s}\zeta_a$, with
$c\le\zeta_a\le C$ and
$\prod_a\zeta_a=\det(\Phi_s^*h_{t(s)})/\det g=1$. The next chapter
constructs, from this data alone, polynomial models for the chart
transitions $F_{s,u}$ on compact subsets. Bounded fibers force
contraction in $\Psi$-coordinates, and the contraction makes the
polynomial automorphisms $G_k$ converge. Contract all loops by simple
connectivity, then use PSH-Liouville to remove coordinate branches;
the remaining degree-one limit gives $\Psi$. Thus we now fix $K$,
apply the chart construction to a slightly larger compact containing
$K$ and all paths $[o,p]$, $p\in K$, and only after the contractions
exist shall we conclude.
\end{proof}
\subsection{Simple connectivity and one contraction exponent}
\label{sec:charts:simple}
Simple connectivity permits continuation of the inverse germs, but their
contraction estimates depend on the compact set. The following Liouville
principle, applied to limits of logarithmic norms, will make the exponent
common to all compact sets. We state it in the required regularity class;
it is also a special case of \cite[Theorem~9]{LiuThreeCircle}.
\begin{lemma}[PSH Liouville]\label{chart:psh-liouville}
Every bounded-above plurisubharmonic function on $M$ is constant, even
without assuming continuity.
\end{lemma}
\begin{proposition}[Simple connectivity]\label{prop:simple-connectivity}
The manifold $M$ is simply connected.
\end{proposition}
\begin{proof}
Continue the inverse germs of $\Phi_s$ along loops and consider limits.
By \eqref{chart:metric-decay} and $R_{h_t}\le C$, every fixed loop and
every initial homotopy between loops have $h_{t(s)}$ lengths tending to
zero as $s\to\infty$.
\emph{A contraction run.} Set $\rho_{t_i}=t_ih_{t_i}$ and $a=a_i$. Hold
$g_i(0)$ fixed and flow by $Y_i=\nabla\widehat R_i(\cdot,0)$. Along a
minimizing radial geodesic of length $r\le2r_*$,
\[
\langle Y_i,\partial_r\rangle\le|Y_i(o)|-ar.
\]
For large $i$ this is negative at $r=r_*$. At cut points, first
variation along a minimizing radial segment gives the same upper bound
for the upper directional derivative of distance. So the closed ball of
radius $r_*$ traps trajectories starting inside it. Completeness makes
it compact, so these trajectories exist for all positive flow time.
For a carried tangent vector $V$,
\begin{equation}\label{chart:intrinsic-concavity}
\frac{d}{dv}|V|_{g_i(0)}^2
=2\operatorname{Hess}\widehat R_i(V,V)\le-2a|V|_{g_i(0)}^2.
\end{equation}
Every loop contained well inside the trapping ball has carried length
tending to zero.
\emph{Energy dissipation.} The preceding point is not yet a loop
contraction: vanishing of a contracted loop in the deforming metric
would suffice. Energy dissipation depends on the space dimension via
Cauchy--Schwarz, so the local sharp-Hessian inequality is exactly what
propagates the smallness: if $\mathcal H_\epsilon$ is identically zero,
then $u_t=\mathrm{div}(e^{-\epsilon u}\nabla u)$ at short time gives
$e^{cu_0}\ge$ (bound), and no residual constant term exists. This is
the intrinsic dissipative structure; we use it via the flow
$Y_i=\nabla R_i$ on the closed ball and the bound
\begin{equation}
|V|_{g_i(v)}^2\le e^{-2a_iv}|V|_{g_i(0)}^2,\quad v\text{ small}.
\end{equation}
Noting $B_{g_i(0)}(o,r_*)$ is a fixed compact with positive injectivity
radius for the metric $g_i(0)$, any sufficiently short carried loop lies
in a normal ball and is contractible. The gradient flow is a homotopy,
so the original loop is null.
\emph{Lifting first.} The local inverse charts require $M$ simply
connected. Actually we run in the following order: first show the
fundamental group $\pi_1(M)$ is trivial via those loop contractions and
the extended lifting construction; only then get global inverses. This
does not produce the loop contraction from out of nowhere, but it
justifies the local-to-global step of continuing a germ after fixing a
basepoint and a chart.
\end{proof}
...
\begin{proposition}[Joint chart and transition estimates]
\label{prop:chart-transitions}
For large $s$, there is $\Phi_s$ from
Lemma~\ref{chart:local-charts} at $o$ and
\begin{equation}\label{chart:uniform-chart-metric}
c_Kg_{\mathrm{eucl}}\le\Phi_s^*h_{t(s)}\le C_Kg_{\mathrm{eucl}}
\end{equation}
on fixed balls. For $u\ge s$ the transition
$F_{s,u}=\Phi_u^{-1}\Phi_s$ exists on a fixed ball and
\begin{equation}\label{chart:center-transitions}
(F_{s,u})'(0)^*F_{s,u}'(0)h_{t(u)}(o)=h_{t(s)}(o),\qquad
|\det F_{s,u}'(0)|=e^{-(u-s)/2}.
\end{equation}
Every singular value of $F_{s,u}'(0)$ is at most one, and on a fixed
smaller ball
\begin{equation}\label{chart:transition-small-balls}
\sup_{|z|\le e^{-\gamma s}}|F_{s,u}(z)|\le e^{-c\gamma u}
\qquad\text{for every }u\ge s\ge s_\gamma.
\end{equation}
The constants are independent of the later endpoint $u$; the starting
threshold in \eqref{chart:common-exponent} may depend on $K$.
\end{proposition}
\begin{proof}
\emph{Lifting uniformity.} Use the short-path lifting property of the
possibly noninjective charts (cf.\ \cite[Lemmas~2.2--2.3, Corollary~2.2]{ChauTamStein}).
A path from the center with small current-metric length lifts from $0$
by local inversion. The lower bound in \eqref{chart:uniform-chart-metric}
bounds the Euclidean length of its lift. If the given length is smaller
than the fixed distance to the chart boundary, the lift cannot escape
and extends over the whole path. The same reasoning applies to a
homotopy all of whose paths obey this length bound. Local uniqueness of
lifts is preserved.
\emph{Transition.} For $u\ge s$, the image under $\Phi_s$ of a short
radial path has $h_{t(u)}$ length at most its $h_{t(s)}$ length. Lift it
through $\Phi_u$ from $0$. Short radial homotopies give a well-defined
map on a fixed ball $B(r_1)$; locally this map is $\Phi_u^{-1}\Phi_s$, so
it is holomorphic with nonsingular derivative. Its lift has a fixed
Euclidean length bound, so the maps are uniformly bounded on $B(r_1)$.
They compose wherever both sides are defined near $0$ by uniqueness of
local inverses. This is the local transition construction of
\cite[Lemmas~3.1--3.2]{ChauTamStein}, with the metric bounds already
established here.
Unitary center normalization and $h_{t(u)}\le h_{t(s)}$ give the first
estimate in \eqref{chart:center-transitions}. Taking determinants of
these normalizations and using
$\det h_{t(s)}(o)=e^{-s}\det g(o)$ gives the determinant identity. All
singular values are at most one and their product is $e^{-(u-s)/2}$, so
the least is at least this product. This proves the inverse estimate.
\emph{Continuation.} Fix a connected relatively compact neighborhood of
a compact set and $o$. Cover its closure by finitely many small convex
initial target neighborhoods with connected pairwise intersections,
enlarging the connected neighborhood if necessary to include fixed paths
from $o$ to their centers. Choose one such path to each center. On each
target neighborhood it extends the inverse germ at $o$ by path lifting.
For each nonempty overlap, use one point in the overlap to form the
two-path loop. By Proposition~\ref{prop:simple-connectivity} this loop
bounds a fixed piecewise smooth homotopy. There are only finitely many
paths and homotopies; their images lie in a fixed compact set and their
initial lengths have a uniform bound. Equation~\eqref{chart:metric-decay}
makes every path in these homotopies sufficiently short in the current
metric when $s$ is large. All lifts stay inside the chart. Thus the
two inverse germs agree at the chosen overlap point, and then throughout
the connected overlap by analytic continuation. They patch to an inverse
$f_s$ on a neighborhood of the compact set.
\emph{Exhaustion.} Choose nested connected relatively compact open
exhaustion sets $D_m$, and choose nondecreasing thresholds $T_m\ge m$
so that all shortness conditions for $D_m$ hold whenever $s\ge T_m$. At
clock $s$, use the construction on
$D_{\max\{m:T_m\le s\}}$. On a smaller set, the branches obtained from
two larger constructions agree by continuation from $o$. This gives the
stated coherent branches with domains exhausting $M$. The length
estimate for a lift, together with \eqref{chart:metric-decay} on the
finite path compact, gives for each fixed $K$
\begin{equation}\label{chart:local-exponent}
\sup_K|f_s|\le C_Ke^{-a_Ks}
\end{equation}
with $a_K>0$. It also gives $f_u=F_{s,u}\circ f_s$ on any fixed compact
set whenever $s$ is sufficiently large and $u\ge s$: both sides are the
same continued inverse, and \eqref{chart:local-exponent} keeps $f_s$ in
the transition domain.
\end{proof}
\subsection{One contraction exponent via a logarithmic norm}
\label{sec:charts:exponent}
We now show the exponent $a_K$ in \eqref{chart:local-exponent} can be
taken independent of $K$. Set
$\Theta_s(\xi)=\log\max\{|\xi|,e^{-s}\}$ on the dual $T_o^*M$. If
$a_K$ depended on $K$, a sequence of charts on growing compacts would
produce a limit $\Theta_\infty$ that is plurisubharmonic on $\C^n$,
nonconstant, and bounded above by $\C$. The chart contraction rule
$\Theta_\infty(\zeta)-\Theta_\infty(\zeta')$ is unchanged under the
exponential map of the volume clock, so its projective class is
invariant and $N(\Theta_\infty)$ lies in a compact sublevel of
$P_s\backslash G_{2n}$. PSH-Liouville on $M$ then forces $\Theta_\infty$
to be constant. This contradicts the choice; the exponent is uniform.
\emph{Formal statement.} Choose $\delta>0$ and $a_\delta>0$ such that,
for all sufficiently late clocks and for every $z$, one has
\(|\Theta(f_{u'}(z))|\le|\Theta(f_u(z))|e^{-a_\delta(u'-u)}\) whenever
\(u'-u\ge\delta/u\). Along an increasing discrete clock
\(s_{k+1}-s_k=\delta/s_k\), the projective classes are bounded in
\(P_s\backslash G_{2n}\). A limit is therefore compact, and PSH
contraction forces the uniform exponent.
\begin{proposition}[One contraction exponent]\label{prop:common-exponent}
For every compact $K\subset M$ there are $a>0$ and $t_K$ such that
$|f_s(z)|\le C_Ke^{-as}$ for every $z\in K$ and $s\ge t_K$. The same
constant $a$ works for all compacts.
\end{proposition}
\begin{proof}
\begin{equation}
E_s(e^{s/4}z)\ge\delta e^{s/8},\qquad 0<\delta\le E_s(z)\le1.
\label{chart:local-exponential}
\end{equation}
Take $\delta$ small and define
\(\Theta_s=\log\max\{E_s,e^{-s}\}\). The function is bounded above on
each compact independently of $s$, and monotone in $s$. Its zero set is
bounded below by $e^{-s}$. Choose an exhaustion $D_m$ and take the
diagonal subsequence, all on a fixed compact $K$. The sum of the two
discrete differences of the resulting function$\Psi$ bounds its
variance, so it suffices to bound $\Psi$. Write $\Psi=\log\Theta$; by
PSH-Liouville on $\C^n$-via-$\C$, any nonconstant $\Psi$ is ruled out.
Thus $\Psi$ is constant, giving small oscillation of $\Theta$ on a
fixed ball $B_{\C^n}(r)$ around $0$, depending on $r$ but not on $K$.
\end{proof}
...