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% Uniformization of complete Kahler manifolds with positive bisectional
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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}
The joint positivity of the cotangent norm will now give 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:
we will 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 this Section, a metric on a cotangent space is the Hermitian
dual metric, including the transpose in its matrix representation.
\subsection{From joint positivity to completeness}
The differential Harnack inequalities for K\"ahler--Ricci flow have
their classical antecedent in Cao's work \cite{CaoHarnack}.
Here we derive the needed inequalities from joint positivity before
using them to prove completeness of the evolving metric.
\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+A h_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 $(x,w)$, where $t=e^{w+\bar w}$, is
plurisubharmonic. Completing the square in the free first derivative
of a cotangent section identifies this assertion with Griffiths
nonnegativity of the pulled-back tangent metric. At a spatially
K\"ahler-normal point its curvature blocks are
\[
\Theta_{i\bar j\,\alpha\bar\beta}
=\Rm_{i\bar j\alpha\bar\beta},\qquad
\Theta_{w\bar j\,\alpha\bar\beta}
=t\nabla_{\bar j}A_{\alpha\bar\beta},\qquad
\Theta_{w\bar w}=tA+t^2(A_t+A h_t^{-1}A).
\]
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
\[
\tr_{h_t}A_t=R_t-|A|_{h_t}^2,
\qquad \tr_{h_t}(A h_t^{-1}A)=|A|_{h_t}^2,
\qquad \tr_{h_t}\nabla_X A=\partial_XR
\]
give \eqref{chart:scalar-harnack}.
At a fixed point choose a moving unitary frame satisfying
$E_t=\tfrac12 A^{\#}E$, where $A^{\#}=h_t^{-1}A$.
Differentiation shows that
\[
\frac{d}{dt}\bigl[tA(E,\overline E)\bigr]
=t\bigl(A_t+A h_t^{-1}A+t^{-1}A\bigr)(E,\overline E)\ge0.
\]
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 this positivity
and gives $tA\ge a_Kh_t$ there for $t\ge1$, after changing the compact
constant. Integrating $\partial_t h_t=-A$ 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.
For clarity, the next integration precedes the completeness proof.
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 the 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 then letting
$\eps\downarrow0$ proves \eqref{chart:integrated-harnack} without
using a minimizing geodesic or completeness of $h_t$.
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}
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.
There is a minimizing segment realizing $l_L(t)$, contained in
$\overline D_L$, and 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 the summed real index forms use perpendicular
parallel fields times a cutoff which rises from zero to one on
an endpoint interval of width
\[
a=\min\left(\frac l2,\frac{\sqrt t}{1+l}\right).
\]
If $l\ge2\sqrt t/(1+l)$, the index inequality and the two endpoint
curvature bounds imply
\[
\int_0^l\Ric_{h_t}(\dot\gamma,\dot\gamma)\,dr
\le \frac C a+\frac{C_T(1+l^2)a}{t}
\le\frac{C_T(1+l)}{\sqrt t}.
\]
Here and below fixed real-versus-complex normalization factors are
included 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 $c_Kg_{\rm eucl}\le\Phi^*h_t\le C_Kg_{\rm eucl}$.
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 $o$.
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.
Here is why no injectivity-radius hypothesis enters the chart
statement; 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 $\bar\partial_J^*\bar\partial_J f=0$,
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. 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 $u_j$
and their equations as $\dbar_{J_j}u_j=\eta_j$.
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 each $u_j$ is qualitatively smooth.
For 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 this equation 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. The qualitative smoothness just
proved legitimizes this test. Uniform metric equivalence gives a
$W^{1,2}$ bound in the fixed real coordinates. Interior elliptic
estimates \cite[Theorems~8.8 and~8.10]{GilbargTrudinger}, on
successively smaller fixed balls, now yield
\[
\|u_j\|_{H^{m+2}(B')}
\le C_m\bigl(\|u_j\|_{L^2(B)}+\|F_j\|_{H^m(B)}\bigr).
\]
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.
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 the 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{Total distortion in the initial metric}
In an $h_t$-unitary chart at $x$, the product of the initial-metric
singular lengths is $e^{\rho(x,t)/2}$. This determinant identity alone
allows the growth to occur in only some directions. The next lemma
bounds the greatest singular length by a fixed power of the least,
so volume loss will force contraction in every direction.
\begin{lemma}[Static distortion]\label{lem:static-distortion}
Fix $r,c>0$ and a compact set $K\subset M$. Suppose
$\Phi:B_{\C^n}(r)\to M$ is a holomorphic local biholomorphism,
$\Phi(0)\in K$, and $\widetilde g=\Phi^*g\ge c g_{\rm eucl}$.
Let $A$ and $B$ be the least and greatest singular lengths of
$d\Phi(0)$, using the Euclidean metric in the domain and $g$ in
the target. There are constants depending only on $K,r,c$ and
the initial geometry on a compact neighborhood of $K$ such that
\begin{equation}\label{chart:static-distortion}
B\le C(1+A)^C.
\end{equation}
In particular these constants are independent of the particular
chart and its total distortion.
\end{lemma}
\begin{proof}
We construct a plurisubharmonic weight whose upper bound is of order
$\log(1+A)$ and whose Levi form is strict near the center.
Its moving-ball definition will also give a nonnegative lower bound
on the support of the weighted $\dbar$ construction. These bounds
make the extension estimate independent of the total distortion.
We then extend a local function whose differential detects $B$;
its $L^2$ bound gives the required polynomial bound.
Choose a constant holomorphic covector $\alpha=df$ with
$|\alpha|_{\rm eucl}=A$ and $|\alpha|_{\widetilde g}(0)=1$.
The logarithmic dual norm
\[
\psi=\log|\alpha|_{\widetilde g}^2
\quad\text{satisfies}\quad
\psi(0)=0,\qquad \psi\le C+2\log A.
\]
For a tangent $X$, the curvature formula for this logarithmic norm is
\[
\dd\psi(X,\overline X)
=\frac{\Rm(X,\overline X,v,\overline v)}{|v|^2}
+\frac{|D'_X\alpha|^2|\alpha|^2
-|\langle D'_X\alpha,\alpha\rangle|^2}{|\alpha|^4},
\]
where $v$ is the metric dual of $\alpha$ and all norms in this
formula use $\widetilde g$. The second summand is nonnegative.
Initial bisectional positivity therefore makes $\psi$
plurisubharmonic. On any region whose target image
lies in a prescribed compact neighborhood $K^+$ of $K$, it gives
\begin{equation}\label{chart:strict-covector}
\dd\psi\ge\kappa\widetilde g
\end{equation}
with $\kappa>0$ uniform. The normalization of $\alpha$ cancels
from this estimate.
Shrink the Euclidean domain by a fixed factor. Choose $r_0>0$
so small that any path of $\widetilde g$ length at most $3r_0$
starting in this smaller domain stays in $B(r)$; the lower
metric bound makes this choice uniform. The small closed metric
balls used below are compact there and their distances are
attained by minimizing segments. Put
\begin{equation}\label{chart:metric-ball-envelope}
\Psi(z)=\max_{\dist_{\widetilde g}(z,y)\le r_0}\psi(y).
\end{equation}
This function is continuous. Compactness gives upper
semicontinuity. For lower semicontinuity, move a maximizing
endpoint a little towards the center along a minimizing segment;
the resulting endpoint is admissible for all nearby centers and
its value tends to the maximum.
We verify the Levi inequality for this moving-ball maximum.
Let $\varphi$ be a smooth upper test at $z$, let $y$ realize the
maximum, and prescribe a nonzero complex tangent $V$ at $z$.
Choose a holomorphic center germ with this derivative. If
$\dist_{\widetilde g}(z,y)<r_0$, any holomorphic endpoint germ
with derivative equal to the parallel translate of $V$ along a
minimizing segment stays admissible for a sufficiently small
parameter. Suppose instead that the constraint is active.
For two real parameter directions corresponding to $V$ and $JV$,
choose trial paths whose first variation fields are the parallel
translates along this segment. Their first length variations
vanish. The trace of their second length variations is
\[
-\int_0^{r_0}
\bigl[\Rm(T,V^{\parallel},V^{\parallel},T)
+\Rm(T,JV^{\parallel},JV^{\parallel},T)\bigr]\,dr<0.
\]
Tangential components make no curvature contribution, and
holomorphic endpoint germs have zero trace covariant
acceleration, so there are no omitted trace boundary terms.
The strict sign is precisely the initial bisectional sign.
The trace inequality alone is insufficient to preserve a
two-parameter length constraint. At the far endpoint we may
prescribe its holomorphic second jet freely while retaining
its first derivative. Changing that jet by a real tangent $W$
(regarded as a complex tangent by $J$) changes the real length
Hessian by
\[
\begin{pmatrix}
\langle W,T\rangle&\langle JW,T\rangle\\
\langle JW,T\rangle&-\langle W,T\rangle
\end{pmatrix}
\]
at the final endpoint. Vectors in the real span of $T,JT$
realize every trace-free symmetric matrix of size two.
Choose the jet to remove the trace-free part of the length
quadratic. The trace remains
strictly negative; hence the full quadratic is negative definite.
Smooth trial paths joining these holomorphic endpoint germs
then have length less than $r_0$ for every sufficiently small
nonzero parameter. The actual distance is no greater than this
trial length. This proves admissibility also at an active
constraint, without differentiating a possibly nonsmooth
distance function.
On either kind of admissible endpoint germ $y(a)$,
$\varphi(z(a))\ge\Psi(z(a))\ge\psi(y(a))$, with equality at
$a=0$. Taking the parameter Levi derivative proves
$\dd\Psi\ge0$ in upper-test sense, and thus in the
plurisubharmonic sense. If
$\dist_{\widetilde g}(0,z)<r_0/4$, all the relevant paths and
endpoints map into a fixed compact $K^+$; applying
\eqref{chart:strict-covector} to the transported tangent gives
\begin{equation}\label{chart:strict-envelope}
\dd\Psi\ge\kappa\widetilde g
\quad\text{on }B_{\widetilde g}(0,r_0/4),
\qquad \Psi\ge0\quad\text{on }B_{\widetilde g}(0,r_0).
\end{equation}
The latter inequality uses the admissible endpoint $0$.
Throughout the fixed Euclidean domain,
$\Psi\le C+2\log A$.
We next specify the weighted extension, including its support.
A fixed small initial holomorphic coordinate neighborhood $U$
of $p=\Phi(0)$ lifts to a single sheet through $0$. Indeed its
short radial paths and their short homotopies lift by local
inversion of $\Phi$, and their $\widetilde g$ lengths equal
their initial target lengths. The lower metric bound prevents
escape from the coordinate domain. Uniqueness of local lifts
and the short homotopies make this an inverse branch on $U$.
Choose $U$ uniformly over $p\in K$, with the whole sheet inside
$B_{\widetilde g}(0,r_0/4)$.
In these initial target coordinates $w$, centered at $p$, choose
a linear holomorphic function $\ell$ whose differential has
target norm one and whose pullback differential at $0$ has
Euclidean norm $B$. Let $\chi$ be a fixed cutoff on this sheet,
equal to one near $0$ and zero near its boundary. The function
$a=\chi\ell$, extended by zero outside the sheet, is smooth and
uniformly bounded. Its source $\beta=\dbar a$ is supported in
a fixed target coordinate annulus and satisfies
\[
|\beta|_{\widetilde g^{-1}}\le C,
\qquad
\int_{\supp\beta}dV_{\rm eucl}
\le c^{-n}\int_{\supp\beta}dV_{\widetilde g}
\le C.
\]
The last integral is the volume of one initial target annulus,
since the chosen sheet is injective. These constants do not
contain $B$.
On the same sheet choose a nonpositive logarithmic-pole weight
$\lambda$, equal to $(n+1)\log(|w|^2/\delta^2)$ near $0$ and
cut off to zero within the strict region. Its negative Hessian
is bounded by $C\widetilde g$ on the cutoff annulus, uniformly
also for smooth pole regularizations. Choose a fixed large
$k$ so that the weight
\[
\varphi_*=k\Psi+\lambda+|z|^2
\]
is plurisubharmonic and its Levi form dominates a positive
multiple of $\widetilde g$ on $\supp\beta$.
On that support, $\Psi\ge0$ and $\lambda$ is bounded below.
The scalar weighted $\dbar$ estimate on the fixed Euclidean
ball \cite{Hormander}\cite[Theorem~5.1 and Remark~4.5]{Demailly} gives
$\dbar v=\beta$ with
\begin{equation}\label{chart:weighted-extension}
\int |v|^2e^{-\varphi_*}\,dV_{\rm eucl}
\le\int |\beta|^2_{(\dd\varphi_*)^{-1}}
e^{-\varphi_*}\,dV_{\rm eucl}\le C.
\end{equation}
For a smooth-weight justification, first convolve $\Psi$ on
a slightly larger Euclidean region and regularize the pole.
For each particular smooth $\widetilde g$, the convolution
scale can be chosen small enough to retain half of
\eqref{chart:strict-envelope} on the compact support regions.
The estimates just obtained are independent of that scale.
Weak $L^2$ limits and Fatou's lemma give
\eqref{chart:weighted-extension} for the singular weight.
The correction $v$ is holomorphic near $0$. Integrability with
the pole $|w|^{-2(n+1)}$ forces its constant and linear terms
to vanish. Thus $a-v$ is holomorphic on the Euclidean ball
and has the prescribed differential of norm $B$ at $0$.
Since $\lambda\le0$ and
$\sup\Psi\le C+2\log A$, dropping the weight in
\eqref{chart:weighted-extension} gives
\[
\|a-v\|_{L^2(dV_{\rm eucl})}^2\le C(1+A)^{2k}.
\]
Interior differentiation of a holomorphic function bounds its
center derivative by this norm, proving
\eqref{chart:static-distortion}.
\end{proof}
\subsection{The volume clock and selected Ricci windows}
The letter $s$ now denotes the volume clock, rather than the time
variable used in the disc envelope. Set
\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 will govern the chart transitions and their polynomial
models. We will also select late times at which the rescaled Ricci
tensor is uniformly pinched at $o$ and the scalar Harnack inequality
approaches equality. The surrounding intervals provide compactness
for the limits used to contract loops.
\begin{proposition}\label{prop:ricci-windows}
The function $q$ is nondecreasing, $q\ge c>0$ for large $s$, and
\begin{equation}\label{chart:q-growth}
s\longrightarrow\infty,\qquad
q\le Ce^\tau,\qquad
\liminf_{s\to\infty}\frac{q(s)}s\le2.
\end{equation}
Uniformly on every fixed compact set, $\rho(x,t)/s\to1$, and
there is $c_K>0$ such that, eventually on that compact set,
\begin{equation}\label{chart:metric-decay}
e^{-2s}g\le h_t\le e^{-c_Ks}g.
\end{equation}
There are fixed $a_0\in(0,1)$ and $b_0>0$ such that the least
eigenvalue at $o$ of $t\Ric(h_t)$ relative to $h_t$ satisfies
\begin{equation}\label{chart:ricci-lower}
\lambda_{\min}(s)\ge b_0q(a_0s)
\quad\text{for all sufficiently large }s.
\end{equation}
Moreover there are $S_i\to\infty$ and $K_0<\infty$ with
\begin{equation}\label{chart:good-windows}
q(2S_i)\le K_0q(a_0S_i).
\end{equation}
One can choose $s_i\in[5S_i/4,7S_i/4]$ so that
$(\log q)'(s_i)\to0$. If $t_i$ corresponds to $s_i$ and
$R_i=R(o,t_i)$, the rescaled flows
\begin{equation}\label{chart:rescaled-flows}
g_i(z)=R_i h_{t_i+z/R_i}
\end{equation}
have uniform curvature and derivative bounds on each fixed-radius
moving ball about $o$ in a fixed small two-sided time interval.
At $(o,0)$ their scalar curvature is one and their Ricci
eigenvalues have a fixed positive lower bound.
\end{proposition}
\begin{proof}
Proposition~\ref{prop:harnack-completeness} gives monotonicity and
the positive lower bound for $q$, together with $q\le Ce^\tau$.
Hence $s\to\infty$. If $q(s)>2s$ for every sufficiently large
$s$, integration of $ds/d\tau>2s$ would give
$s\ge ce^{2\tau}$, contradicting the linear-in-$t$ bound for
$\rho(o,t)$. This proves \eqref{chart:q-growth}.
For a compact family of initial paths from $o$, the power
contraction in Proposition~\ref{prop:harnack-completeness} makes
their $h_t$ lengths tend uniformly to zero. The factor on the
right of \eqref{chart:integrated-harnack}, with time increment
one, therefore tends uniformly to one on the compact set. Apply
the inequality in both spatial directions and integrate in time.
More explicitly, for every $\eps>0$ and all sufficiently large
$v$, uniformly on the compact set,
\[
R(x,v)\le(1+\eps)R(o,v+1),\qquad
R(o,v)\le(1+\eps)R(x,v+1).
\]
The first inequality also bounds $R(x,v)$ there. Shifting either
integral endpoint by one costs a bounded amount. After division
by $s\to\infty$, and then letting $\eps\downarrow0$, the two
inequalities give $\rho(x,t)/s\to1$. No integrability of the
error factor is required.
Let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $h_t$
relative to $g$ at $x$. They lie in $(0,1]$, and their product
is $e^{-\rho(x,t)}$. Thus every $\lambda_j\ge e^{-\rho(x,t)}$,
which proves the lower bound of \eqref{chart:metric-decay}.
Choose the instantaneous unitary chart of
Lemma~\ref{chart:local-charts} at $x$. Its initial-metric
singular lengths are $\lambda_j^{-1/2}$, whose product is
$e^{\rho(x,t)/2}$. Lemma~\ref{lem:static-distortion} bounds
their largest value by a fixed power of their smallest, uniformly
for $x$ in the compact set. Since all these lengths are at least
one, their product forces the smallest to be at least
$C^{-1}e^{c\rho(x,t)}$. This proves the upper bound of
\eqref{chart:metric-decay}, after reducing $c_K$ and increasing
the starting time.
For each fixed nonzero covector $\xi\in T_o^{*(1,0)}M$, the
function
\[
L_\xi(\tau)=\log|\xi|_{h_{e^\tau}^{-1}}^2
\]
is convex. Indeed the logarithmic norm of a holomorphic dual
section is plurisubharmonic for a Griffiths-nonnegative tangent
metric, and its restriction to complexified $\tau$ is independent
of the imaginary part. Its derivative is the Rayleigh quotient
of $t\Ric$ on the dual unit vector. Normalize $|\xi|_{g^{-1}}=1$.
At $o$, \eqref{chart:metric-decay} gives
\[
L_\xi(\tau(s))-L_\xi(\tau(a_0s))
\ge(c_o-2a_0)s.
\]
Choose $a_0<c_o/4$, reducing it further to be less than one if
needed. Monotonicity of $q$ gives
\[
\tau(s)-\tau(a_0s)
=\int_{a_0s}^s\frac{dv}{q(v)}
\le\frac{s}{q(a_0s)}.
\]
The endpoint derivative of a convex function dominates its secant
slope. Taking the minimum over all covectors proves
\eqref{chart:ricci-lower}.
The estimates obtained so far hold at every sufficiently late clock.
We now select intervals on which the lower Ricci bound is comparable
to the scalar curvature. Put $L=2/a_0>1$ and choose $K_0>L^2$. If
\eqref{chart:good-windows} failed for all large $S$, then
$q(Lr)>K_0q(r)$ for all large $r$. Iteration and monotonicity
would give $q(r)\ge cr^{\log K_0/\log L}$ for every large $r$,
contradicting \eqref{chart:q-growth}. Thus such windows occur
arbitrarily far out. On their middle halves,
\[
\int_{5S_i/4}^{7S_i/4}(\log q)'(s)\,ds\le\log K_0.
\]
Choose $s_i$ there with
$(\log q)'(s_i)\le2\log K_0/S_i$.
Write $q_i=q(s_i)=t_iR_i$. On the rescaled clock of
\eqref{chart:rescaled-flows},
\begin{equation}\label{chart:clock-rescaling}
\frac{ds}{dz}=\frac{q(s)}{q_i+z},\qquad
\widehat R_i(o,z)=\frac{R(o,t_i+z/R_i)}{R_i}
=\frac{q(s)}{q_i+z}.
\end{equation}
Since $q_i\ge c>0$, choose $\delta<c/4$. As long as $s$ lies
in the good window, $|ds/dz|\le2K_0$ for $|z|\le\delta$.
The chosen $s_i$ lie a distance at least $S_i/4$ from its ends,
so this estimate keeps the whole fixed time interval in the
window for large $i$. Formula~\eqref{chart:clock-rescaling}
then bounds the rescaled basepoint scalar curvature above and
below by fixed positive constants. Apply the rescaled
\eqref{chart:integrated-harnack}, using a fixed future-time
buffer, to get a uniform scalar bound on every fixed-radius
moving ball in a smaller interval. Bisectional nonnegativity
controls $|\Rm|$, and Lemma~\ref{chart:local-charts} supplies
the local higher bounds. Finally
\eqref{chart:ricci-lower} and \eqref{chart:good-windows} give
at the central point
\[
\lambda_{\min}(\Ric(g_i))
\ge\frac{b_0q(a_0s_i)}{q_i}\ge\frac{b_0}{K_0},
\qquad \widehat R_i(o,0)=1.
\]
\end{proof}
\subsection{Harnack equality and contraction of loops}
The uniform charts are still only local biholomorphisms. To continue
their inverse germs to single-valued maps on compact subsets of $M$,
we must remove possible monodromy. The selected windows do this by
producing equality in a limiting Harnack inequality. We will show
that this equality forces real strict concavity of the scalar
curvature, then use its gradient flow to contract loops in the
corresponding small intrinsic balls.
\begin{proposition}\label{prop:simple-connectivity}
The manifold $M$ is simply connected.
\end{proposition}
\begin{proof}
Take the rescaled flows in Proposition~\ref{prop:ricci-windows}.
Their pointed pullbacks on tangent balls have a smooth local
subsequential limit, denoted $g(z)$. Pass to a further subsequence
so that
\[
k_i(z)=\frac1{q_i+z}\longrightarrow k(z),\qquad k_z=-k^2.
\]
This includes $k=0$ when $q_i\to\infty$. Denote the limit
scalar curvature by $R$ and its Ricci form by $A$. At the
central spacetime point, $R=1$ and $A\ge b\,g$ with $b>0$.
Formula~\eqref{chart:clock-rescaling} gives the exact identity
\[
\left[\partial_z\widehat R_i+k_i\widehat R_i\right]_{(o,0)}
=(\log q)'(s_i)\longrightarrow0.
\]
The limit therefore has $S:=R_z+kR=0$ at the central point.
The matrix Harnack inequality has the rescaled form
$A_z+A g^{-1}A+kA\ge0$. Its trace is $S$, because
$R_z=\tr_g A_z+|A|_g^2$. Thus this nonnegative matrix vanishes
at the point in question. Also $A_z=\dd R$ under K\"ahler--Ricci
flow. If $H_{j\bar l}=\nabla_j\nabla_{\bar l}R$, then
\begin{equation}\label{chart:equality-mixed-hessian}
H=-A^2-kA
\end{equation}
there, writing endomorphism products in a unitary frame.
On a small spacetime neighborhood, $A$ is positive definite.
Minimizing the scalar Harnack inequality over $X$ gives
\begin{equation}\label{chart:equality-Z}
Z=S-Q\ge0,\qquad Q=|\partial R|_{A^{-1}}^2.
\end{equation}
At the equality point $S=0$, so $\partial R=0$ and $Z=0$.
We give the scalar Hessian calculation, since a vanishing
Harnack quantity by itself does not supply real strict concavity.
Use $R_z=\Delta R+|A|^2$, $A_z=\dd R$, and $k_z=-k^2$.
The derivative of the Laplacian on a scalar is
$(\partial_z\Delta)R=\langle A,\dd R\rangle$, while
\[
\partial_z|A|^2=2\langle A,\dd R\rangle+2\tr(A^3).
\]
Consequently
\begin{equation}\label{chart:equality-evolution}
(\partial_z-\Delta)S
=3\langle A,\dd R\rangle+2\tr(A^3)
+k|A|^2-k^2R.
\end{equation}
At the equality point, substitution of
\eqref{chart:equality-mixed-hessian} reduces this to
$-\tr\bigl(A(A+k\Id)^2\bigr)$.
Diagonalize $A$ there, with eigenvalues $\lambda_j>0$.
Since $\partial R=0$, derivatives of the coefficients of
$A^{-1}$ make no contribution to $\Delta Q$ or $Q_z$, and
\[
Q_z=0,\qquad
\Delta Q=\sum_{j,l}\lambda_j^{-1}
\left(|R_{j\bar l}|^2+|R_{jl}|^2\right).
\]
The first square sum is exactly
$\tr\bigl(A(A+k\Id)^2\bigr)$ by
\eqref{chart:equality-mixed-hessian}. Hence at this point
\[
(\partial_z-\Delta)Z
=\sum_{j,l}\lambda_j^{-1}|R_{jl}|^2\ge0.
\]
On the other hand $Z\ge0$ has an interior spacetime zero, so
$Z_z=0$ and $\Delta Z\ge0$ there. The displayed sum must
vanish. Thus the pure complex Hessian $R_{jl}$ is zero, while
the mixed Hessian is negative definite by
\eqref{chart:equality-mixed-hessian}. The real Hessian of $R$
is therefore negative definite at the center, with a definite
bound furnished by the positive lower bound for $A$.
This conclusion holds uniformly on actual small metric balls
in the original rescaled manifolds, not only on their immersed
parametrizations. Smooth convergence gives the center Hessian
bound and $|\nabla\widehat R_i|(o,0)\to0$. The intrinsic
third-derivative estimates from
Proposition~\ref{prop:ricci-windows} allow one to parallel
transport the Hessian along any minimizing radial geodesic.
Shrinking a fixed radius $r_*>0$, independently of $i$, gives
constants $a>0$ with
\begin{equation}\label{chart:intrinsic-concavity}
\operatorname{Hess}_{g_i(0)}\widehat R_i
\le-a g_i(0)
\quad\text{on }B_{g_i(0)}(o,2r_*),\qquad
|\nabla\widehat R_i|(o,0)\longrightarrow0.
\end{equation}
Indeed the change in the transported Hessian is bounded by
$Cr_*$, so one chooses $r_*$ after the uniform center bound.
This argument uses intrinsic derivative estimates along actual
geodesics and is unaffected by multiple sheets of a local chart.
Hold the complete metric $g_i(0)$ fixed and flow by the real
vector field $Y_i=\nabla\widehat R_i(\cdot,0)$. 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 on $r=r_*$. At cut points,
first variation along any minimizing radial segment gives the
same upper bound for the upper directional derivative of
distance. Thus the closed ball of radius $r_*$ traps trajectories
starting inside it. Completeness makes this closed ball compact,
so these trajectories exist for all positive flow time.
For a carried tangent vector $V$, \eqref{chart:intrinsic-concavity}
gives
\[
\frac{d}{dv}|V|_{g_i(0)}^2
=2\operatorname{Hess}\widehat R_i(V,V)
\le-2a|V|_{g_i(0)}^2.
\]
Every loop contained well inside the trapping ball consequently
has carried length tending exponentially to zero.
Let a fixed piecewise smooth loop in $M$ be given. Its image
and a finite collection of paths from $o$ lie in a fixed compact
set. Their $h_{t_i}$ lengths tend to zero by
\eqref{chart:metric-decay}, and $R_i\le C$ by
Proposition~\ref{prop:harnack-completeness}. Thus this loop
lies inside $B_{g_i(0)}(o,r_*/4)$ for sufficiently large $i$.
Fix one such $i$. The compact trapping ball has a positive
minimum injectivity radius for this one smooth metric. A
sufficiently short carried loop lies in a normal ball and is
contractible. The gradient flow up to that finite time is a
homotopy from the original loop to it. Approximation of
continuous loops by piecewise smooth loops finishes the proof.
\end{proof}
\subsection{Inverse charts and one contraction exponent}
Simple connectivity permits continuation of the inverse germs, but
their first contraction estimates will 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}\label{chart:psh-liouville}
Every bounded-above plurisubharmonic function on $M$ is constant,
including a function which is not assumed continuous.
\end{lemma}
\begin{proof}
Allow the constant $-\infty$, and otherwise let $v$ be such a
function. The compact-ball maximum
$m(r)=\max_{\overline B_g(o,r)}v$ is finite and nondecreasing.
It is convex as a function of $\log r$. To see this, fix
$0<r_1<r_2$ and compare on the closed annulus with the
logarithmic chord
\[
H(r)=m(r_1)
+\frac{m(r_2)-m(r_1)}{\log r_2-\log r_1}
(\log r-\log r_1).
\]
If its slope is zero the desired comparison is immediate from
the definition of $m(r_2)$. For positive slope, a positive
maximum of $v-H(\dist_g(o,\cdot))$ would occur in the annulus
interior. At such a point choose a minimizing radial segment.
The length index form with linear parallel test fields in the
$J$-radial direction gives an upper Hessian bound for distance.
More explicitly, let $T$ be the final radial unit vector, let
$T(\ell)$ denote its parallel extension, and let $\bar r$ be
the length upper support obtained from these trial paths. At
the endpoint of a segment of length $r$ the real Hessian satisfies
\[
\operatorname{Hess}\log\bar r(T,T)
+\operatorname{Hess}\log\bar r(JT,JT)
\le-\frac1{r^3}\int_0^r
\ell^2\Rm(T(\ell),JT(\ell),JT(\ell),T(\ell))\,d\ell<0.
\]
Indeed the radial term is $-1/r^2$, while the linear test field
in the $JT$ direction gives $1/r^2$ minus the displayed
curvature integral. This is the strictly negative Levi
derivative on the radial complex line.
This support argument applies also at cut points: smoothly vary
the endpoint and use trial paths along the chosen minimizing
segment. Their lengths give a smooth upper support with the
same index-form bound. Since $H$ is increasing, it produces
a smooth upper test for $v$ at the hypothetical positive
maximum, with a negative Levi derivative. A plurisubharmonic
function admits no such upper test, whether or not it is
continuous. Hence $v\le H$ on the annulus, proving the
claimed convexity. A nondecreasing convex function of
$\log r\in\R$ which is bounded above must be constant.
Upper semicontinuity implies
$\lim_{r\downarrow0}m(r)=v(o)$, so $v$ attains its global
maximum at $o$. The plurisubharmonic maximum principle on
connected $M$ shows that $v$ is constant.
\end{proof}
Write $J_dF$ for the Taylor jet of $F$ through total degree $d$ at zero,
and fix a coefficient norm on each finite-dimensional jet space.
\begin{proposition}\label{prop:shrinking-charts}
For every sufficiently large volume clock $s$, there is a
holomorphic local biholomorphism
$\Phi_s:B_{\C^n}(r_0)\to M$ centered at $o$, with unitary
center derivative for the metric at clock $s$, and
\begin{equation}\label{chart:uniform-chart-metric}
C^{-1}g_{\rm eucl}\le\Phi_s^*h_{t(s)}\le Cg_{\rm eucl}.
\end{equation}
For fixed $0<r_1<r_0$, chosen sufficiently small, the transition
germs $F_{s,u}=\Phi_u^{-1}\Phi_s$, $u\ge s$, extend to
nonsingular holomorphic maps on $B(r_1)$ with a common bound.
They fix $0$, compose as germs, and satisfy
\begin{equation}\label{chart:center-transitions}
\|F_{s,u}'(0)\|\le1,\qquad
|\det F_{s,u}'(0)|=e^{-(u-s)/2},\qquad
\|F_{s,u}'(0)^{-1}\|\le e^{(u-s)/2}.
\end{equation}
There are coherent inverse branches $f_s$ on neighborhoods of
any fixed compact exhaustion set for all sufficiently large
$s$, satisfying $f_s(o)=0$ and $\Phi_sf_s=\Id$. A constant
$\sigma>0$, common to all compact sets, satisfies
\begin{equation}\label{chart:common-exponent}
\sup_K|f_s|\le e^{-\sigma s}
\quad\text{for every fixed compact }K
\text{ and all sufficiently large }s.
\end{equation}
For every fixed jet order $d$ there is $C_d$ such that
\begin{equation}\label{chart:transition-jets}
|J_dF_{s,u}|\le C_d\min\{1,e^{C_ds-\sigma u}\}
\qquad(u\ge s\text{ sufficiently large}).
\end{equation}
There are $c,\gamma_0>0$ such that, for each fixed
$0<\gamma\le\gamma_0$, one has
\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 in the last two estimates are independent of the
later endpoint $u$; the starting threshold in
\eqref{chart:common-exponent} may depend on $K$.
\end{proposition}
\begin{proof}
Take the charts of Lemma~\ref{chart:local-charts} at $o$.
We use the elementary short-path lifting property of these
possibly noninjective charts; compare
\cite[Lemmas~2.2--2.3 and Corollary~2.2]{ChauTamStein}.
A path beginning at the center and having sufficiently 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 hence extends over the whole path.
The same reasoning applies to a homotopy all of whose paths
obey this length bound. Local uniqueness of lifts supplies
uniqueness throughout the homotopy.
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, and
therefore 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.
We now use simple connectivity to continue the inverse germs.
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.
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.
It remains to make the exponent independent of the compact.
The functions $v_s=s^{-1}\log|f_s|$ are plurisubharmonic on
their exhausting domains and locally bounded above.
For any sequence $s\to\infty$,
Lemma~\ref{lem:psh-compactness-bridge} and diagonal extraction
give either convergence to
$-\infty$ locally uniformly from above, or local $L^1$
convergence to a plurisubharmonic function on $M$.
In the second case the upper-regularized limit is at most
zero everywhere by \eqref{chart:local-exponent}.
Lemma~\ref{chart:psh-liouville} makes it constant. On one
fixed neighborhood of $o$, \eqref{chart:local-exponent}
bounds this constant by a fixed negative number, say
$-a_*$. The constant-limit conclusion of
Lemma~\ref{lem:psh-compactness-bridge} then gives
$\limsup\sup_Kv_s\le-a_*$ on every fixed compact along
this subsequence. The collapsing case gives the same
conclusion with $-\infty$. A sequence violating
\eqref{chart:common-exponent}, with $\sigma=a_*/2$, would
contradict these alternatives. This proves a common
exponent on the full family of clocks.
For the two remaining transition estimates, transfer the common
exponent from a fixed initial neighborhood back to a small ball in
the chart domain. Choose a fixed compact initial neighborhood $K_0$
of $o$ with positive initial distance from $o$ to its
boundary. Until a path $\Phi_s(rz)$ first exits $K_0$,
\eqref{chart:metric-decay} and
\eqref{chart:uniform-chart-metric} bound its initial length
by $Ce^s|z|$. It follows that, for a fixed $C_0>1$ and
all large $s$,
\[
\Phi_s\bigl(B(e^{-C_0s})\bigr)\subset K_0.
\]
For every $u\ge s$, the map $f_u\Phi_s$ on this ball is
the same local inverse composition as $F_{s,u}$ near zero,
and hence everywhere there. Equation~\eqref{chart:common-exponent}
therefore gives the bound
\begin{equation}\label{chart:transition-inner-bound}
\sup_{B(e^{-C_0s})}|F_{s,u}|\le e^{-\sigma u}
\qquad\text{for all }u\ge s\text{ and large }s.
\end{equation}
Cauchy's estimates on this inner ball give the factor
$e^{dC_0s-\sigma u}$ for derivatives of order at most $d$.
Cauchy's estimates on a fixed transition ball give a uniform
bound. Combining the two proves
\eqref{chart:transition-jets}.
For completeness, fix an outer radius $R<r_1$ and a common
bound $B_*$ for the transitions there. Euclidean
three-circle interpolation, applied to their scalar linear
projections and then taking the supremum, gives at radius
$e^{-\gamma s}$, for fixed small $\gamma>0$,
\[
\log\sup_{B(e^{-\gamma s})}|F_{s,u}|
\le-\theta_s\sigma u+(1-\theta_s)\log B_*,
\qquad
\theta_s=\frac{\log(R/e^{-\gamma s})}
{\log(R/e^{-C_0s})}
\ge\frac{\gamma}{2C_0}
\]
for all sufficiently large $s$. In this formula choose
$\gamma_0<C_0$, so the middle radius lies between the
inner and outer radii. Since $u\ge s$, the fixed term
$\log B_*$ is absorbed uniformly for every later endpoint,
giving \eqref{chart:transition-small-balls} with, for
example, $c=\sigma/(4C_0)$ and a threshold depending on
$\gamma$. This proves all the stated uniformities.
\end{proof}
\section{Polynomial models with nonuniform contraction}
\label{sec:dynamics}
We retain the volume clock $s=\rho(o,t)$, its logarithmic-time speed
$q(s)=ds/d\tau=tR(o,t)$, and the charts of
Proposition~\ref{prop:shrinking-charts}.
Polynomial normalization and inverse iteration are classical tools
for attracting basins, notably in Rosay--Rudin's work
\cite[Appendix]{RosayRudin1988}; Chau--Tam brought this strategy into
K\"ahler uniformization \cite{ChauTamComplex2006,ChauTamStein,ChauTamSimply}.
The estimates proved here address the nonuniform contraction permitted
by Proposition~\ref{prop:ricci-windows}. For sequences of holomorphic
automorphisms of $\C^n$, $n\ge2$, fixing zero and satisfying
$A|z|\le|f_j(z)|\le B|z|$ on one fixed ball about zero, with
$0<A<B<1$ independent of $j$, Bera--Verma prove that the attracting
basin is biholomorphic to $\C^n$ \cite[Theorem~1.1]{BeraVerma2024}.
The chart transitions here require the variable-rate estimates below.
In particular, we use only
\begin{equation}
q\text{ nondecreasing},\qquad q\ge c>0,\qquad
\liminf_{s\to\infty}\frac{q(s)}s\le2,\qquad
\lambda_{\min}(t\Ric)(o)\ge b_0q(a_0s),
\label{eq:dyn:rate-input}
\end{equation}
where $0<a_0<1$ and $b_0>0$ are fixed.
We continue to use $J_d$ and the fixed coefficient norms introduced in
Section~\ref{sec:charts}.
All maps and jets in this Section fix zero. A coefficient bound
$\exp(o(s_k))$ means an upper bound $\exp(\eta_k s_k)$ with
$\eta_k\to0$; its rate of convergence may depend on parameters already
fixed. No assertion of uniformity as those parameters vary is intended.
For polynomial automorphisms put
$G_{k,l}=G_{l-1}\circ\cdots\circ G_k$ and $G_{k,k}=\Id$.
\begin{proposition}[Polynomial models]\label{prop:polynomial-models}
Fix an integer $d\ge2$ and a macro width $0<h<1$. After starting at a
sufficiently late macro endpoint, there is a grid $s_k\to\infty$ with
macro endpoints $S_j=e^{hj}$ and the following properties. Each macro
interval is either one bad step or is divided into regular steps of
length comparable to $\sqrt{S_j}$. Thus, eventually,
\begin{equation}
c_h\sqrt{s_k}\le\Delta s_k:=s_{k+1}-s_k
\le(e^h-1)s_k.
\label{eq:dyn:grid-size}
\end{equation}
For $F_k=F_{s_k,s_{k+1}}$ there are polynomial local coordinate changes
$H_k$ and polynomial automorphisms $G_k$ of $\C^n$ satisfying
\begin{equation}
J_d(H_{k+1}\circ F_k)=J_d(G_k\circ H_k).
\label{eq:dyn:jet-conjugacy}
\end{equation}
The degrees of $H_k,G_k,G_k^{-1}$ are bounded independently of $k$.
The coefficients of $H_k$ and its inverse jet are $\exp(o(s_k))$;
$H_k'(0)$ and its inverse have polynomial bounds in $s_k$.
For a constant $C_d$ depending only on the fixed jet order and dimension,
\begin{align}
\norm{G_k}_{\mathrm{coeff}}+\norm{G_k^{-1}}_{\mathrm{coeff}}
&\le \exp\bigl(C_d\Delta s_k+o(s_k)\bigr),
\label{eq:dyn:coefficient-bound}\\
\norm{G_k'(0)^{-1}}
&\le s_k^{C}\exp(\Delta s_k/2).
\label{eq:dyn:center-inverse}
\end{align}
In particular, the coefficient $C_d$ of $\Delta s_k$ is independent of
$h$. Finally, for some finite $D$,
\begin{equation}
\deg G_{0,k}\le D s_k
\quad\text{on an unbounded subsequence of macro endpoints.}
\label{eq:dyn:forward-degree}
\end{equation}
\end{proposition}
First we normalize the Jacobian jets coherently along the grid.
On regular steps we then construct polynomial models whose nonlinear
terms and intervening linear changes preserve bounds for the degree of
each component. On bad steps, realization of the full normalized jet
costs a fixed degree factor. The last part of the proof charges these
factors to growth of $q$ and obtains \eqref{eq:dyn:forward-degree}.
The distinction between the constant $C_d$ in
\eqref{eq:dyn:coefficient-bound} and constants hidden in $o(s_k)$ will
be used in Section~\ref{sec:uniformization}.
\subsection{Coherent volume jets}
On the uniform transition ball, the germs $F_k$ are nonsingular and have
uniformly bounded derivatives of each fixed order on a smaller ball.
Their center derivatives are contractions, and chart normalization gives
\begin{equation}
\abs{\det F_k'(0)}=e^{-\Delta s_k/2},\qquad
\norm{F_k'(0)^{-1}}\le e^{\Delta s_k/2}.
\label{eq:dyn:transition-linear}
\end{equation}
The second inequality follows because every singular value is at most
one and their product is $e^{-\Delta s_k/2}$.
Choose the holomorphic logarithm vanishing at zero of
\[
P_k(z)=\log\frac{\det F_k'(z)}{\det F_k'(0)}.
\]
The upper bound on its real part is $C+\Delta s_k/2$. The coefficient
estimate for a holomorphic function with bounded real part, applied on
concentric balls and with $P_k(0)=0$, therefore gives
$\norm{J_{d-1}P_k}\le C_d(1+\Delta s_k)$.
The volume correction uses only the spacing bounds
\eqref{eq:dyn:grid-size}, so it can be constructed before the macro
intervals are classified. On any such grid the jet series
\begin{equation}
\ell_k=\sum_{l\ge k}J_{d-1}
\bigl(P_l\circ F_{s_k,s_l}\bigr)
\label{eq:dyn:volume-series}
\end{equation}
converges. Indeed Proposition~\ref{prop:shrinking-charts} bounds each
coefficient of the summand by
\[
C_d(1+\Delta s_l)
\min\{1,\exp(C_d s_k-\sigma s_l)\}.
\]
The absence of a constant term in $P_l$ is essential here. For
$s_l\le C'_d s_k$ the step lengths telescope and the number of steps is
polynomial in $s_k$. Beyond that range the displayed exponential is
summable, since the grid eventually has spacing at least one and
$\Delta s_l\le(e^h-1)s_l$. Thus $\ell_k$ has polynomial coefficient
bounds in $s_k$. The composition identity for transitions yields
\begin{equation}
\ell_k=J_{d-1}\bigl(P_k+\ell_{k+1}\circ F_k\bigr).
\label{eq:dyn:volume-cocycle}
\end{equation}
Let $E_k=J_{d-1}\exp(\ell_k)$ and define a polynomial change
\[
A_k(z)=\left(z_1,\ldots,z_{n-1},
\int_0^{z_n}E_k(z_1,\ldots,z_{n-1},\zeta)\,d\zeta\right).
\]
It is tangent to the identity, has degree at most $d$, and satisfies
$J_{d-1}\log\det A_k'=\ell_k$. Its coefficients and inverse jet have
polynomial bounds in $s_k$. In the new coordinates the normalized
logarithmic Jacobian jet of $A_{k+1}F_kA_k^{-1}$ is
\[
J_{d-1}\bigl(\ell_{k+1}\circ F_k+P_k-\ell_k\bigr)\circ A_k^{-1}=0.
\]
This records the sign of the volume correction explicitly. We shall
call a jet with Jacobian determinant equal, through degree $d-1$, to
its center determinant a \emph{constant-volume jet}.
\begin{lemma}[Realization of constant-volume jets]\label{lem:volume-jets}
For fixed $n,d$, every invertible constant-volume $d$-jet fixing zero
is the $d$-jet of a polynomial automorphism of $\C^n$. Its degree and
inverse degree are bounded by a constant $D_d$. Its coefficients and
inverse coefficients are bounded polynomially in the coefficients of
the given jet and its inverse linear part. These bounds do not depend
on the grid or on $h$.
\end{lemma}
\begin{proof}
Normalize the derivative to the identity. If a volume-preserving
polynomial automorphism agrees with the required jet through degree
$m-1$, the first discrepancy is a homogeneous vector polynomial $X_m$
of degree $m$ with $\operatorname{div}X_m=0$: this is the degree $m-1$
part of the Jacobian determinant condition.
The space of such fields is spanned by
\begin{equation}
X(z)=l(z)^m v,\qquad l\in(\C^n)^*,\quad v\in\C^n,\quad l(v)=0.
\label{eq:dyn:shear-fields}
\end{equation}
This is the divergence-free shear lemma of Anders\'en
\cite{AndersenVolume1990}; see also
\cite[proof of Theorem~3.1]{ForstnericLarusson}.
We include the finite-dimensional proof to track the coefficient bounds.
Pair a linear functional on
homogeneous vector polynomials with $l^m v$. The result has the form
$A(l)(v)$, where $A(l)$ is a covector-valued homogeneous polynomial of
degree $m$. If the functional annihilates all fields in
\eqref{eq:dyn:shear-fields}, then $A(l)$ is proportional to $l$.
The polynomial identities $l_iA_j(l)=l_jA_i(l)$ show that
$A(l)=B(l)l$ for a homogeneous polynomial $B$ of degree $m-1$.
Such functionals are exactly the image of the transpose of divergence,
since $\operatorname{div}(l^m v)=m l(v)l^{m-1}$. Taking annihilators
proves the spanning assertion. In dimension one the divergence kernel
in these degrees is zero, so this argument also covers $n=1$.
Select a finite spanning list of the fields
\eqref{eq:dyn:shear-fields}, once for each $2\le m\le d$.
The time-$c$ map of such a field is the polynomial shear
$z\mapsto z+c l(z)^m v$, whose inverse has the minus sign; both have
Jacobian determinant one. A fixed linear decomposition of $X_m$
therefore gives a finite composition of shears correcting degree $m$
without changing lower degrees. Induction through $m=d$, followed by
the original linear map, proves realization. There are a fixed number
of shears, and each operation on the truncated jets is polynomial in
the preceding coefficients and the inverse linear part. This proves
all the coefficient and degree bounds, including those for the inverse.
Restoring the original linear map gives the realizing automorphism
the determinant of that linear map; the correcting shears have
determinant one.
\end{proof}
\subsection{Rate order without commuting matrices}
Consider one interval in logarithmic time, of length $T>0$.
Singular-value decompositions in its start and end unitary tangent
coordinates give derivative
\begin{equation}
D=\diag(e^{-\mu_1T/2},\ldots,e^{-\mu_nT/2}),\qquad
0\le\mu_1\le\cdots\le\mu_n,\qquad
\sum_i\mu_i=\frac{\Delta s}{T}.
\label{eq:dyn:diagonal-rates}
\end{equation}
The volume changes $A_k$ do not change these linear parts.
We need an order statement for the rate forms, and not merely their
ordered eigenvalues. Normalize the cometric at the start to $I$ in a
fixed covector basis. In the diagonalized basis its value at the end
is $\diag(e^{\mu_iT})$. On the complex strip $0\le\Re z\le T$, the
holomorphic covector with components $a_i e^{-\mu_i z/2}$ has squared
norm $\abs a^2$ on both boundary lines. Its squared norm is bounded
and subharmonic by Theorem~\ref{thm:joint-positivity}, applied with
$t=t_{\mathrm{start}}e^{\Re z}$; equivalently the complex time variable
there is shifted by $z/2$. The bounded subharmonic maximum principle
on the strip gives the same upper bound inside. Consequently the
cometric is bounded above by the flat exponential interpolation of its
endpoint values. Differentiating this matrix inequality at the two
ends gives
\begin{equation}
(t\Ric)_{\mathrm{start}}\le\diag(\mu_i),\qquad
\diag(\mu_i)\le(t\Ric)_{\mathrm{end}},
\label{eq:dyn:endpoint-rate-order}
\end{equation}
where each matrix is expressed in the corresponding endpoint unitary
covector coordinates. The inequalities have opposite endpoint signs
because the matrix difference vanishes at both ends and is
nonnegative inside.
At a shared endpoint of two steps, let $U_k$ send old tangent
components to new tangent components. It is unitary; a row covector
$v$ in the new coordinates has old row $vU_k$. Combining the two
inequalities in \eqref{eq:dyn:endpoint-rate-order} gives
\begin{equation}
\sum_l\mu_l^{\mathrm{old}}\abs{(vU_k)_l}^2
\le\sum_i\mu_i^{\mathrm{new}}\abs{v_i}^2.
\label{eq:dyn:loewner-jump}
\end{equation}
This proof allows the instantaneous Ricci matrices to fail to commute.
In particular, the sorted rates are nondecreasing from step to step.
\subsection{The macro grid and propagated weights}
The positive weights below will bound ordinary polynomial degrees
component by component. For a polynomial map
$P=(P_1,\ldots,P_n):\C^n\to\C^n$ with $\deg P_l\le w_l$, one has
\[
\deg(P_1^{\beta_1}\cdots P_n^{\beta_n})
\le\sum_l\beta_lw_l.
\]
Thus a nonlinear map preserves these degree bounds if every monomial
$z^\beta$ in component $i$ satisfies
$\sum_l\beta_lw_l\le w_i$. A linear change between two weight systems
preserves the bounds provided every nonzero entry sends an old
coordinate to a new coordinate of at least its weight. We shall
enforce both rules on regular steps. Bad steps may multiply degrees
by a fixed factor; their defining rate growth will pay for that factor.
Put $Q_j=q(S_j)$ and choose an integer $N\ge1$ with
$e^{-hN}\le a_0$. A macro index $j$ is \emph{flagged} when
\begin{equation}
\log Q_{j+1}-\log Q_{j-N}>M,
\label{eq:dyn:flag}
\end{equation}
where the fixed threshold $M$ will be chosen sufficiently large below.
Declare bad the union of the expanded flag intervals
$[j-N,j+1+N)$ in the integer macro index line. Each bad macro is a
single step. Divide every remaining macro interval into equal steps
whose lengths are comparable to $\sqrt{S_j}$, for example by taking
the number of pieces to be the nearest larger integer to
$(S_{j+1}-S_j)/\sqrt{S_j}$. Once $j$ is large this gives
\eqref{eq:dyn:grid-size}.
If macro $j$ is regular, it is not flagged. On each of its steps,
\eqref{eq:dyn:rate-input} and \eqref{eq:dyn:endpoint-rate-order} imply
\begin{equation}
b_0Q_{j-N}\le\mu_i\le Q_{j+1},\qquad
bQ_j\le\mu_i\le b^{-1}Q_j
\label{eq:dyn:regular-rates}
\end{equation}
for a fixed $b>0$ depending on $M,b_0$. These constants may depend on
$h$ after all parameter choices have been made.
For a regular macro set $\theta_j=Q_j/j^2$ and choose bounded increasing
multipliers $\lambda_j$ on the whole macro sequence by
\[
\lambda_{j+1}/\lambda_j=1+K_0/j^2.
\]
Their product is bounded, since $\sum j^{-2}<\infty$; an overall
constant factor is free. Choose $0<\epsilon_0<1/(10n)$.
For each regular macro, form a finite directed graph whose vertices
are the pairs $(k,i)$ consisting of a step in that macro and one of
its coordinates. Give this vertex raw weight
$\lambda_j(\mu_i-\theta_j)$, where $\mu_i$ is its rate in step $k$.
Draw an edge to a vertex in the same or the next step precisely when
the destination rate is at least the source rate minus
$\epsilon_0\theta_j$. Define $w_{k,i}$ as the maximum raw weight
among vertices from which $(k,i)$ is reachable, allowing paths of
length zero. This maximum exists because the graph is finite, even
when same-step edges form cycles. Assign the weight to both ends of
its diagonal step.
This propagation loses at most $n\epsilon_0\theta_j$ in rates,
independently of the number of steps. To verify the assertion, follow
any permitted chain and keep its successive lower records. Once a
sorted index has appeared, its rate at a later occurrence is at least
its earlier rate, by \eqref{eq:dyn:loewner-jump}. A new lower record
can therefore occur only on the first visit to an index; each such
record lowers the previous record by at most
$\epsilon_0\theta_j$. There are at most $n$ indices. It follows that
\begin{equation}
\lambda_j(\mu_i-\theta_j)\le w_{k,i}
\le\lambda_j(\mu_i-0.9\theta_j).
\label{eq:dyn:weights}
\end{equation}
In particular, whenever a permitted passage is possible the weights
do not decrease.
The same nondecrease can be required across adjacent regular macros.
Indeed, write $\mu'$ for a new rate and $\mu$ for an old one, with
$\mu'\ge\mu-\epsilon_0\theta_j$. A lower bound on the new raw weight
minus the upper bound on the old propagated weight is
\[
(\lambda_{j+1}-\lambda_j)\mu'
-\lambda_{j+1}\theta_{j+1}
+(0.9-\epsilon_0)\lambda_j\theta_j.
\]
Since $\mu'\ge bQ_{j+1}$, choosing $K_0$ large in terms of $b$ makes
this expression nonnegative at every sufficiently large index.
Thus the passage rule is valid also at regular macro junctions.
Increase the overall scale of $\lambda$ so that all weights are at
least one after the eventual starting index.
Call a nonlinear monomial $z^\beta$ in component $i$ \emph{allowed}
when $\sum_l\beta_lw_{k,l}\le w_{k,i}$, and \emph{forbidden} otherwise.
For a forbidden monomial of degree $m\ge2$,
\eqref{eq:dyn:weights} gives
\begin{equation}
\sum_l\beta_l\mu_l-\mu_i
>(0.9m-1)\theta_j\ge0.8\theta_j.
\label{eq:dyn:forbidden-gap}
\end{equation}
For an allowed monomial, every variable occurring in it has strictly
smaller weight than its output: the weights are positive and $m\ge2$.
Thus allowed nonlinear maps are triangular after ordering coordinates
by weight.
\subsection{Aligning the linear jumps}
\begin{lemma}[Linear alignment]\label{lem:linear-alignment}
On every run of regular steps there are unipotent matrices $L_k^+$,
with off-diagonal entries only in rows of smaller weight than their
columns, such that, on putting $L_k^-=D_k^{-1}L_k^+D_k$, the diagonal
step derivative remains $D_k$ and every regular jump becomes
\begin{equation}
J_k=L_{k+1}^-U_k(L_k^+)^{-1},\qquad
(J_k)_{il}=0\quad\hbox{if }w_{k+1,i}<w_{k,l}.
\label{eq:dyn:aligned-jump}
\end{equation}
The matrices $L_k^+$ and their inverses have polynomial bounds in the
macro index $j$, while
\begin{equation}
\norm{L_k^- -I}
\le j^C\exp\left(-c\frac{\sqrt{S_j}}{j^2}\right).
\label{eq:dyn:small-start-change}
\end{equation}
The estimates are uniform in finite terminal horizons, and hold on an
infinite regular run by a coefficientwise limit.
\end{lemma}
\begin{proof}
Work backward, taking $L_k^+=I$ at the last step of a finite run or
horizon. Suppose the next small start adjustment is known and set
$B=L_{k+1}^-U_k$. For a real threshold $x$, let $A_x$ be the span of
rows of $B$ whose new weights are at most $x$, and let $E_x$ be the old
coordinate row space with weights at most $x$.
Projection $P_{E_x}:A_x\to E_x$ is injective, and its inverse on its
image has norm $O(j)$. For the proof assume $A_x\ne0$ and let $r_x$
be the largest new rate among its defining rows. Every old rate
outside $E_x$ exceeds $r_x+\epsilon_0\theta_j$: otherwise the passage
rule would make that old weight at most the weight of a selected new
row. For any unit $a\in A_x$, \eqref{eq:dyn:loewner-jump} and the
small next adjustment imply that its old rate Rayleigh quotient is at
most $r_x+o(\theta_j)$. To see the size of this error, express $a$ as
a combination of the selected rows of $L_{k+1}^-U_k$; its coefficient
norm is $1+o(1)$, and the error is bounded by the operator norm of
$L_{k+1}^- -I$ times the largest old or new rate. The regular rate
bounds make that error $o(\theta_j)$, also at a macro junction.
Nonnegativity of all old rates now gives
\[
(r_x+\epsilon_0\theta_j)
(1-\norm{P_{E_x}a}^2)
\le r_x+o(\theta_j).
\]
As $r_x\le C Q_j$, this implies
$\norm{P_{E_x}a}^2\ge c/j^2$. Empty or full coordinate spaces give
the same assertion in its evident form.
We give the graph extension explicitly. List the distinct spaces
$E_x$ increasingly. For each such space retain the largest required
$A_x$ with that $E_x$, so that the required $A_x$ remain nested.
Starting with the full row space, descend through the list. Suppose
the next larger space $V_+$ already projects isomorphically to $E_+$
and contains the current $A=A_x$. Write $T_+:E_+\to V_+$ for its
inverse projection, $E=E_x$, $Y=P_EA$, and $R_A:Y\to A$ for the
inverse of $P_E|_A$. Define
\begin{equation}
T_E(e)=R_A(P_Ye)+T_+((I-P_Y)e),\qquad e\in E.
\label{eq:dyn:graph-extension}
\end{equation}
Here orthogonal projection $P_Y$ is taken in $E$. Both terms lie in
$V_+$, projection to $E$ is $e$, and the range contains $A$.
Moreover $\norm{T_E}\le\norm{R_A}+\norm{T_+}$.
There are at most $n$ nontrivial stages, so the resulting graph lifts
have polynomial bounds in $j$.
For each coordinate row, take its lift at its own weight threshold.
These rows form $L_k^+$. The rows up to every threshold span the
corresponding nested graph: they lie there and their projections form
a triangular basis of its coordinate space. Each lifted row equals
its coordinate row plus terms of strictly higher weight. Hence
$L_k^+=I+N_k$ with $N_k$ nilpotent, and
$(L_k^+)^{-1}=\sum_{r=0}^{n-1}(-N_k)^r$ has polynomial bounds.
The inclusion $A_x(L_k^+)^{-1}\subset E_x$ is exactly the zero pattern
in \eqref{eq:dyn:aligned-jump}.
For every nonzero off-diagonal entry in row $i$, column $l$, the
within-step passage rule gives $\mu_l-\mu_i>\epsilon_0\theta_j$.
Conjugation therefore gives
\[
(D_k^{-1}L_k^+D_k)_{il}
=(L_k^+)_{il}\exp\bigl(- (\mu_l-\mu_i)\Delta\tau_k/2\bigr).
\]
By \eqref{eq:dyn:regular-rates},
$\theta_j\Delta\tau_k\ge c\sqrt{S_j}/j^2$.
This proves \eqref{eq:dyn:small-start-change}; its exponential absorbs
every polynomial bound needed in the preceding backward step.
After a sufficiently late start the induction thus closes uniformly
in the horizon. A diagonal subsequence of finite-horizon matrices
proves the assertion on infinite runs; the zero conditions and all
bounds are closed under that limit.
\end{proof}
\subsection{Homological equations and polynomial automorphisms}
Choose a single primary coordinate system at each grid endpoint.
Use the adjusted start coordinates if a regular step starts there;
otherwise use the adjusted end coordinates of a regular step just
ended, if there is one; use the volume coordinates in all other cases.
These choices are a linear change, of polynomial size together with
its inverse, following $A_k$. For a regular step, before the possible
jump to the next regular start, write the resulting transition as
$\widehat F_k$, with derivative $D_k$. Its actual transition to the
next primary coordinates is $J_k\widehat F_k$, with $J_k=I$ when the
next step is bad. It has a constant-volume $d$-jet and polynomially
bounded coefficients. This follows by formal composition and
inversion of the primary coordinate jets; only their local germs are
used.
We construct tangent-to-identity polynomial jets $h_k$ at regular
starts, taking $h_k=I$ elsewhere. For a regular step seek
$G_k=J_kg_k$, where $g_k$ has linear part $D_k$ and only allowed
nonlinear terms, satisfying
\begin{equation}
g_k\circ h_k
=J_k^{-1}\circ h_{k+1}\circ J_k\circ\widehat F_k
\quad\text{through degree }d.
\label{eq:dyn:homological-identity}
\end{equation}
In homogeneous degree $m$, once lower degrees have been fixed, this
is
\begin{equation}
g_k^{(m)}+D_kh_k^{(m)}
=J_k^{-1}h_{k+1}^{(m)}(J_kD_kz)+B_k^{(m)}(z),
\label{eq:dyn:homological-layer}
\end{equation}
where $B_k^{(m)}$ contains only already known data.
Put only forbidden coefficients into $h_k^{(m)}$ and only allowed
coefficients into $g_k^{(m)}$. If $\Pi_k$ projects to the forbidden
coefficients, the equation for $h_k^{(m)}$ is a backward affine
recursion with linear operator
\begin{equation}
T_kX=\Pi_k\left[D_k^{-1}J_k^{-1}X(J_kD_kz)\right].
\label{eq:dyn:backward-operator}
\end{equation}
Conjugation by $J_k$ has polynomial norm in $j$ at each fixed degree.
After it has been expanded, a coefficient in output $i$ and input
$\beta$ receives the diagonal factor
$\exp[-(\sum_l\beta_l\mu_l-\mu_i)\Delta\tau_k/2]$.
Projection and \eqref{eq:dyn:forbidden-gap} consequently give
\begin{equation}
\norm{T_k}\le j^{C_m}
\exp\left(-c\frac{\sqrt{S_j}}{j^2}\right).
\label{eq:dyn:backward-contraction}
\end{equation}
This estimate holds for every incoming homogeneous polynomial $X$;
no support restriction on $X$ has been imposed before conjugation.
Solve first on a finite regular horizon with terminal correction the
identity, and proceed by degree. Inductively the lower-layer forcing,
including division by $D_k$, has coefficients bounded by
$\exp(C'_m\sqrt{s_k})$: regular steps have
$\Delta s_k\asymp\sqrt{s_k}$, and
$\norm{D_k^{-1}}\le e^{\Delta s_k/2}$.
The envelopes $\exp(C_m\sqrt{s_k})$ have bounded consecutive ratios,
because $\sqrt{s_{k+1}}-\sqrt{s_k}$ is bounded on regular steps.
Thus \eqref{eq:dyn:backward-contraction} is eventually less than
one half after weighting by these envelopes. The backward geometric
series solves \eqref{eq:dyn:homological-layer} with
\begin{equation}
\norm{h_k}+\norm{J_dh_k^{-1}}+\norm{g_k}
+\norm{g_k^{-1}}_{\mathrm{coeff}}
\le\exp(C_d'\sqrt{s_k}),
\label{eq:dyn:regular-coefficients}
\end{equation}
uniformly in the terminal horizon. Here $C_d'$ may depend on all
fixed grid parameters. Taking coefficientwise diagonal limits gives
the same construction on an infinite regular run.
Each $g_k$ is a triangular polynomial automorphism: all nonlinear
inputs have smaller weights than their output. Its inverse is found
by successive substitution in increasing weight order. Its degree
and inverse degree are bounded in terms of $n,d$, and its determinant
is the constant $\det D_k$.
The coordinate jets $h_k$ also preserve volume through degree $d-1$.
This is not a property assumed of the forbidden projection. In a
finite-horizon problem, differentiate the complete jet identity
\eqref{eq:dyn:homological-identity}. The determinant of
$\widehat F_k'$ has the same constant jet as $\det D_k$, and $g_k$ has
that determinant exactly. If $\det h_{k+1}'=1$ through degree $d-1$,
then the determinant identity forces the same statement for $h_k$.
Backward induction from the identity terminal jet proves it, and it
passes to the coefficientwise limits.
At a bad macro step realize, by Lemma~\ref{lem:volume-jets}, the jet of
the transition between the two primary coordinate systems after their
endpoint corrections $h_k,h_{k+1}$. It is a constant-volume jet by
what was just proved. Enlarge $D_d$ to bound the degrees and inverse
degrees of these models and of the regular models.
Let $H_k$ be the degree-$d$ truncation of the total primary change
followed by $h_k$. It has the required center derivative and inverse
jet, and \eqref{eq:dyn:jet-conjugacy} follows.
For precision, on a bad step the realized input jet has coefficient
bound $\exp(o(s_k))$, whereas its inverse linear part is bounded by
$\exp(\Delta s_k/2+o(s_k))$, by
\eqref{eq:dyn:transition-linear}. Lemma~\ref{lem:volume-jets} costs a
fixed polynomial in these two quantities. Its polynomial exponent
depends only on $n,d$. Therefore it gives
\eqref{eq:dyn:coefficient-bound} with a coefficient $C_d$ independent
of $h,N,M$. All endpoint and regular costs
$\exp(C(d,h,N,M)\sqrt{s_k})$ remain in $\exp(o(s_k))$.
Finally the linear part of the jet conjugacy is
$G_k'(0)=H_{k+1}'(0)F_k'(0)H_k'(0)^{-1}$; the polynomial bounds on the
endpoint linear changes give \eqref{eq:dyn:center-inverse}.
\subsection{Charging bad components to rate growth}
\begin{lemma}[Forward degree control]\label{lem:degree-control}
The flag threshold $M$ can be chosen so that regular macro starts
occur arbitrarily late, and the polynomial models constructed above,
initialized at any sufficiently late such start, satisfy
\eqref{eq:dyn:forward-degree}.
\end{lemma}
\begin{proof}
Set
\[
W=2N+1,\qquad D_*=D_d,\qquad
B_* =\max\{0,\log(2/b_0)\},\qquad \alpha=\frac{M}{3W}.
\]
Choose $M$ so large that
\begin{equation}
\alpha\ge\log D_*+B_*/W,\qquad \alpha>h.
\label{eq:dyn:flag-threshold}
\end{equation}
A flag $f$ has expanded interval $I_f=[f-N,f+1+N)$, of length $W$,
and growth interval $[f-N,f+1]$, on which $\log Q$ increases by more
than $M$.
Consider a full finite bad component $[u,v)$. Choose a maximal
disjoint collection of its expanded flag intervals. Their concentric
triples cover the component: any unselected equal-length interval
meets a selected one and is contained in its triple. Thus there are
at least $(v-u)/(3W)$ selected intervals. Their growth intervals are
disjoint up to endpoints and lie in $[u,v-N]$. Monotonicity gives
\begin{equation}
\log\frac{Q_{v-N}}{Q_u}\ge\alpha(v-u).
\label{eq:dyn:full-bad-growth}
\end{equation}
Every full component has length $L=v-u\ge W$, so
\eqref{eq:dyn:flag-threshold} implies
\begin{equation}
D_*^L Q_u\le e^{-B_*L/W}Q_{v-N}
\le(b_0/2)Q_{v-N}.
\label{eq:dyn:bad-reset-cost}
\end{equation}
We now track the ordinary degree of each component of the accumulated
forward polynomial. At a regular start bound these degrees by the
respective weights. Allowed nonlinear terms preserve that bound
under composition, and \eqref{eq:dyn:aligned-jump} preserves it at a
regular jump. If the next macro is bad, retain the last end weights;
at entry to $[u,v)$ their maximum is at most
$\lambda_{u-1}Q_u$. Each bad map multiplies maximum degree by at most
$D_*$. At the next regular macro $v$, its flag test fails, and hence
$Q_v\le e^M Q_{v-N}$. For sufficiently large $v$ every new start
weight is at least
\[
\lambda_v\bigl(b_0Q_{v-N}-Q_v/v^2\bigr)
\ge\lambda_v(b_0/2)Q_{v-N}.
\]
As $\lambda_v\ge\lambda_{u-1}$,
\eqref{eq:dyn:bad-reset-cost} resets all degree bounds exactly to the
new weights. There is no multiplicative loss for a completed bad
component.
Prefixes require a separate estimate. For a bad prefix $[u,j)$,
the expanded flag intervals wholly contained in the prefix cover
$[u,j-W)$, when this interval is nonempty. Indeed an expanded interval
covering a point before $j-W$ ends before $j$, and none crosses the
left endpoint $u$ of the component. The same disjoint selection gives
\begin{equation}
\log(Q_j/Q_u)\ge\alpha(j-u-W)_+,
\qquad D_*^{j-u}Q_u\le D_*^W Q_j.
\label{eq:dyn:bad-prefix-growth}
\end{equation}
This proof also applies to a prefix of an infinite bad component.
Such a component would give
$\log(Q_j/S_j)\ge(\alpha-h)j-O(1)\to\infty$.
It is impossible: choose continuous-clock points $x_r\to\infty$ with
$q(x_r)\le3x_r$, and let $S_{j_r}\le x_r<S_{j_r+1}$.
Then $Q_{j_r}\le3e^h S_{j_r}$, contradicting that growth.
Thus there are regular macro starts arbitrarily far out.
Start at one of them, after every fixed-parameter threshold above,
and label this grid point $k=0$, retaining the original large macro
indices $j$. The accumulated map is initially the identity and its
component degrees are one, so the initial weight bound holds after
the overall scaling already allowed for $\lambda$. Regular
propagation, the exact full-component reset, and the single prefix
loss in \eqref{eq:dyn:bad-prefix-growth} give, at all macro endpoints,
\[
\deg G_{0,k(j)}\le
C\max\{e^M,D_*^W\}Q_j,
\]
where $C$ can be taken to depend on the bounded supremum of the
multipliers. On a regular macro one uses
$Q_{j+1}\le e^M Q_j$; inside a bad component one uses the prefix
estimate. Applying the bound at the endpoints $j_r$ just selected
proves \eqref{eq:dyn:forward-degree}. This uses only a subsequence
with $Q_j=O(S_j)$, never an all-time bound on $q(s)/s$.
\end{proof}
Lemmas~\ref{lem:volume-jets}, \ref{lem:linear-alignment}, and
\ref{lem:degree-control}, together with the homological construction,
complete the proof of Proposition~\ref{prop:polynomial-models}.
\section{Convergence and surjectivity of the coordinates}
\label{sec:uniformization}
We now pass from the polynomial models to a global map. The local
errors must remain small under backward polynomial iteration, and the
subsequential forward degree bound must then rule out a proper image.
We give both steps with their domains and parameter choices.
\subsection{Parameter order and local error estimates}
Decrease the common chart exponent $\sigma$ of
Proposition~\ref{prop:shrinking-charts}, if necessary, so that
$0<\sigma<1$. Choose
\begin{equation}
0<\gamma<\gamma'<\min\{\sigma/2,1\},
\qquad 0<10\nu<\min\{c\gamma,\sigma,\gamma,1\},
\label{eq:unif:decay-parameters}
\end{equation}
with $\gamma$ in the range of its small-ball transition estimate;
decrease its constant $c$ to at most one. Fix the jet order so large
that
\begin{equation}
(d+1)\nu>30.
\label{eq:unif:jet-order}
\end{equation}
Only after this choice fix a sufficiently small $h>0$, as specified
below. Then choose $N,M,b,K_0,\lambda$ as in
Section~\ref{sec:dynamics}, and finally choose a sufficiently late
regular macro start. The estimates in
Proposition~\ref{prop:polynomial-models} hold for this choice.
On a fixed smaller raw transition ball, the jet conjugacy and Cauchy
estimates give
\begin{equation}
\abs{H_{k+1}F_k(z)-G_kH_k(z)}
\le \exp\bigl(A_d\Delta s_k+o(s_k)\bigr)\abs z^{d+1}.
\label{eq:unif:remainder}
\end{equation}
Here $A_d$ is independent of $h$. Indeed the degrees of the
polynomials in this expression are fixed, the $F_k$ are uniformly
bounded on a larger raw ball, and the coefficient estimates of
Proposition~\ref{prop:polynomial-models} bound the two compositions
there. They agree through degree $d$, so the Cauchy remainder has
the displayed factor. This argument uses $H_{k+1}F_k-G_kH_k$
directly; it requires no uniform analytic domain for $H_k^{-1}$.
We shall use two sequences of raw chart coordinates:
\begin{enumerate}[label=(\roman*)]
\item $z_k=f_{s_k}(x)$, uniformly for $x$ in any fixed compact
neighborhood in the exhaustion and all sufficiently large $k$;
\item $z_k=F_{s_a,s_k}(z)$, uniformly for
$\abs z\le e^{-\gamma s_a}$, for any sufficiently large $a$ and
all $k\ge a$.
\end{enumerate}
In the first case the threshold may depend on the compact; in the
second it is uniform in $a$. Continuation and composition of the
transition germs give $z_{k+1}=F_k(z_k)$ in both cases.
The chart estimates and \eqref{eq:unif:decay-parameters} imply,
eventually,
\begin{equation}
\abs{z_k}\le e^{-5\nu s_k},\qquad
\zeta_k:=H_k(z_k),\quad
\abs{\zeta_k}\le e^{-2\nu s_k}.
\label{eq:unif:small-orbits}
\end{equation}
In regime (ii), the case $k=a$ uses $F_{s_a,s_a}=\Id$ and
$\gamma>10\nu$.
Choose $h$ so small that, with $\delta=e^h-1$,
\begin{equation}
\delta<\tfrac1{10},\qquad
A_d\delta<1,\qquad C_d\delta<\nu/4,
\label{eq:unif:macro-choice}
\end{equation}
enlarging $A_d,C_d$ first to cover the finitely many fixed-degree
coefficient estimates used here. By
\eqref{eq:unif:jet-order}--\eqref{eq:unif:small-orbits}, after the
late start the defect satisfies
\begin{equation}
r_k:=\zeta_{k+1}-G_k(\zeta_k),\qquad
\abs{r_k}\le e^{-10s_k}.
\label{eq:unif:orbit-defect}
\end{equation}
All constants dependent on $h,N,M$ multiply $o(s_k)$ in this argument
and are absorbed only by that late start. In particular,
\eqref{eq:unif:macro-choice} does not require choosing $h$ using a
constant which is itself determined by $M$.
The inverse maps have the local Lipschitz bound
\begin{equation}
\Lip\left(G_k^{-1}\bigm|_{B(e^{-\nu s_{k+1}})}\right)
\le e^{\Delta s_k}
\label{eq:unif:inverse-tube}
\end{equation}
at all sufficiently large indices. To check it, the derivative of
$G_k^{-1}$ on this ball is bounded by the sum of
\[
s_k^C e^{\Delta s_k/2}
\quad\hbox{and}\quad
\exp\bigl(C_d\Delta s_k+o(s_k)-\nu s_{k+1}\bigr).
\]
The first term is at most $e^{3\Delta s_k/4}$ eventually, since
$\Delta s_k\ge c_h\sqrt{s_k}$. The second tends to zero by
\eqref{eq:unif:macro-choice}; their sum is at most
$e^{\Delta s_k}$ after increasing the starting index. Integrating
the derivative along segments in this convex ball gives
\eqref{eq:unif:inverse-tube}.
\subsection{Backward approximation inside valid tubes}
\begin{lemma}[Convergence of corrected coordinates]
\label{lem:unif:backward-limit}
For either raw sequence above and every sufficiently large $k$,
\[
\xi_k=\lim_{l\to\infty}G_{k,l}^{-1}\zeta_l
\]
exists locally uniformly in the indicated variables. It is
holomorphic and satisfies
\begin{equation}
\abs{\xi_k-\zeta_k}\le e^{-6s_k},\qquad
\xi_{k+1}=G_k\xi_k.
\label{eq:unif:shadow}
\end{equation}
All applications of \eqref{eq:unif:inverse-tube} take place inside
its stated balls.
\end{lemma}
\begin{proof}
For a terminal index $l$ put $y_l^{(l)}=\zeta_l$ and
$y_i^{(l)}=G_i^{-1}y_{i+1}^{(l)}$ for $i<l$.
Simultaneously establish, by backward induction, that the comparison
segments at step $i$ lie in $B(e^{-\nu s_{i+1}})$ and that
\begin{equation}
\abs{y_k^{(l)}-\zeta_k}
\le \sum_{i=k}^{l-1}
e^{-10s_i}\exp(s_{i+1}-s_k).
\label{eq:unif:finite-error-sum}
\end{equation}
The terminal assertion is immediate. At a backward step compare
$G_i^{-1}y_{i+1}^{(l)}$ with
$G_i^{-1}G_i\zeta_i=\zeta_i$.
Both arguments are close to $\zeta_{i+1}$: the first by the inductive
bound and the second by \eqref{eq:unif:orbit-defect}. Since
$\abs{\zeta_{i+1}}\le e^{-2\nu s_{i+1}}$, they and their joining
segment lie in the required larger tube whenever the error sum has
the bound asserted below. Applying
\eqref{eq:unif:inverse-tube} then gives the next error sum.
Indeed $s_{i+1}\le(1+\delta)s_i$ and the grid eventually has spacing
at least one, so
\[
\sum_{i\ge k}e^{-10s_i+s_{i+1}-s_k}
\le e^{-s_k}\sum_{i\ge k}e^{-(9-\delta)s_i}
\le C e^{-(10-\delta)s_k}\le e^{-6s_k}.
\]
As $\nu<1/10$, this is much smaller than the difference between the
two tube radii. The estimates therefore close the simultaneous
induction without evaluating an inverse outside its proved domain.
For completeness, consecutive terminal approximations are Cauchy
by the same reasoning. At level $l$ their difference is at most
$e^{\Delta s_l}e^{-10s_l}$, and propagation down to $k$ bounds it by
$e^{-10s_l+s_{l+1}-s_k}$. These bounds are summable in $l$.
They are uniform on the compact sets in either regime, so the limits
are holomorphic. The finite identities give
$\xi_{k+1}=G_k\xi_k$, and the infinite error sum proves
\eqref{eq:unif:shadow}.
\end{proof}
\subsection{A coherent injective map on the manifold}
\begin{proposition}[Global coordinates]\label{prop:global-coordinates}
The locally uniform limit on the exhaustion
\begin{equation}
\Psi(x)=\lim_{l\to\infty}G_{0,l}^{-1}H_lf_{s_l}(x)
\label{eq:unif:global-map}
\end{equation}
defines an injective holomorphic map $M\to\C^n$ with $\Psi(o)=0$.
Its image $\Omega$ is open, and $\Psi:M\to\Omega$ is a
biholomorphism. Moreover, on every compact $K\subset\Omega$,
\begin{equation}
\sup_K\abs{G_{0,k}}\le e^{-(\sigma/2)s_k}
\quad\hbox{eventually},
\label{eq:unif:forward-smallness}
\end{equation}
and, at every sufficiently large $k$,
\begin{equation}
G_{0,k}^{-1}\bigl(B(e^{-\gamma's_k})\bigr)\subset\Omega.
\label{eq:unif:inverse-ball-inclusion}
\end{equation}
\end{proposition}
\begin{proof}
On a fixed exhaustion compact choose an index $k$ after which regime
(i) is available. Lemma~\ref{lem:unif:backward-limit} gives $\xi_k$
there, and composing it with the single fixed polynomial
automorphism $G_{0,k}^{-1}$ gives the limit
\eqref{eq:unif:global-map}. On overlaps the inverse charts agree by
continuation, and \eqref{eq:unif:shadow} makes the limits agree.
Thus these maps define a holomorphic $\Psi$ on all of $M$. All maps
fix zero in charts, so $\Psi(o)=0$.
Suppose $\Psi(x)=\Psi(y)$ for two fixed points. For every large $k$,
$G_{0,k}\Psi=\xi_k$ on a compact containing them, and hence
\eqref{eq:unif:shadow} implies
\[
\abs{H_kf_{s_k}(x)-H_kf_{s_k}(y)}\le2e^{-6s_k}.
\]
On the tiny raw-coordinate ball containing these points, write
$H_k(z)=A_kz+E_k(z)$, where $A_k=H_k'(0)$ has conorm at least
$s_k^{-C}$ and $\norm{DE_k}$ is bounded by $\exp(o(s_k))$ times
the radius. Integrating only $DE_k$ along the segment and retaining
the fixed linear term $A_k$ gives
\[
|H_k(z)-H_k(w)|\ge
\bigl(s_k^{-C}-\sup\norm{DE_k}\bigr)|z-w|.
\]
The nonlinear error is eventually less than $s_k^{-C}/2$, so
\begin{equation}
\abs{f_{s_k}(x)-f_{s_k}(y)}\le e^{-5s_k}
\label{eq:unif:raw-separation}
\end{equation}
eventually.
If $x\ne y$, choose a small relatively compact initial metric ball
about $x$ whose closure excludes $y$. Map the straight segment
between the two raw coordinates by $\Phi_{s_k}$, obtaining a path
from $x$ to $y$. The segment stays in the uniform chart ball.
Up to its first exit from the fixed ball about $x$, the compact
metric comparison $e^{-2s_k}g\le h_{t(s_k)}$ from
Proposition~\ref{prop:ricci-windows}, and the uniform chart upper
bound $\Phi_{s_k}^*h_{t(s_k)}\le Cg_{\mathrm{Eucl}}$, bound its
initial-metric length by
$Ce^{s_k}e^{-5s_k}$. This tends to zero and cannot reach the
boundary of that fixed ball. The contradiction proves injectivity.
This argument uses immersed charts only on their indicated sheet;
it requires no global injectivity of $\Phi_{s_k}$.
Nonsingularity follows directly from the same quantitative estimates.
Fix a point and two coordinate neighborhoods $U\Subset V$ whose
closures lie in one compact exhaustion set. The estimates of
regime (i) hold uniformly on $\overline V$ for every sufficiently
large $k$. Differentiating $\Phi_{s_k}f_{s_k}=\Id$, and using
\eqref{chart:metric-decay} and
\eqref{chart:uniform-chart-metric}, gives
\[
|df_{s_k}(v)|\ge c e^{-s_k}|v|_g
\quad\text{on }\overline V.
\]
The coefficient bounds on $H_k$, its polynomially bounded inverse
linear part, and $|f_{s_k}|\le e^{-\sigma s_k}$ therefore imply
\[
|d(H_kf_{s_k})(v)|\ge c s_k^{-C}e^{-s_k}|v|_g.
\]
On the other hand, the shadow bound
$|\xi_k-H_kf_{s_k}|\le e^{-6s_k}$ on $\overline V$ and Cauchy's
estimate on the fixed pair $U\Subset V$ bound its differential on
$U$ by $C e^{-6s_k}|v|_g$. Thus
\[
|d\xi_k(v)|\ge
\bigl(c s_k^{-C}e^{-s_k}-C e^{-6s_k}\bigr)|v|_g>0
\qquad(v\ne0)
\]
for all sufficiently large $k$. Since $\xi_k=G_{0,k}\Psi$ and
$G_{0,k}$ is a polynomial automorphism, $d\Psi$ is nonsingular.
The holomorphic inverse theorem now makes the image $\Omega$ open;
the local inverses agree by the proved injectivity, giving a
holomorphic inverse on $\Omega$. Consequently
$K'=\Psi^{-1}(K)$ is compact whenever $K\subset\Omega$ is compact. On $K'$, the common chart bound
$\abs{f_{s_k}}\le e^{-\sigma s_k}$, the coefficient bound on $H_k$,
and \eqref{eq:unif:shadow} give
\[
\sup_K\abs{G_{0,k}}
\le \exp\bigl(-\sigma s_k+o(s_k)\bigr)+e^{-6s_k},
\]
which proves \eqref{eq:unif:forward-smallness}.
It remains to prove the ball inclusion. Fix a sufficiently large
$k$ and use regime (ii) with $a=k$. For large $l$, the inverse chart
$f_{s_l}$ is defined on the compact image under $\Phi_{s_k}$ of the
closed ball $\overline B(e^{-\gamma s_k})$. The identity
$f_{s_l}\Phi_{s_k}=F_{s_k,s_l}$ holds first as a germ at zero and
then throughout that ball by holomorphic continuation. Therefore
Lemma~\ref{lem:unif:backward-limit} gives
\begin{equation}
\sup_{\abs z\le e^{-\gamma s_k}}
\abs{G_{0,k}\Psi\Phi_{s_k}(z)-H_k(z)}
\le e^{-6s_k}.
\label{eq:unif:chart-shadow}
\end{equation}
Write $A=H_k'(0)$, $r=e^{-\gamma s_k}$, and
\[
T_k=G_{0,k}\Psi\Phi_{s_k},\qquad E_k(z)=T_k(z)-Az.
\]
The map $T_k$ is holomorphic on the fixed chart ball, and fixes zero.
Its shadow estimate and the coefficient bound on $H_k$ give
\[
E_k(0)=0,\qquad \norm{A^{-1}}\le s_k^C,\qquad
\sup_{\abs z\le r}|E_k(z)|
\le \exp(o(s_k))r^2+e^{-6s_k}.
\]
Cauchy's estimate on the smaller ball then gives
\[
\sup_{\abs z\le r/2}\norm{A^{-1}DE_k(z)}
\le C_n s_k^C\bigl(\exp(o(s_k))r+e^{-6s_k}/r\bigr)
\longrightarrow0.
\]
For large $k$ this supremum is at most $1/4$. Moreover,
$\gamma'>\gamma$ implies
$|A^{-1}w|\le r/4$ whenever $|w|\le e^{-\gamma's_k}$,
after increasing the starting index once more. For such a target
$w$, the map
\[
z\longmapsto A^{-1}w-A^{-1}E_k(z)
\]
is $1/4$-Lipschitz on $\overline B(r/2)$ and maps this closed ball
into $B(3r/8)$, because $E_k(0)=0$. Its successive iterates are
Cauchy, stay in the closed ball, and converge to a fixed point $z$.
The fixed-point identity is $T_k(z)=w$. Therefore
$B(e^{-\gamma's_k})\subset G_{0,k}(\Omega)$, which is precisely
\eqref{eq:unif:inverse-ball-inclusion}.
\end{proof}
\subsection{The image is all of affine space}
\label{sec:surjectivity}
We first record the polynomial growth estimate used in the final step.
For a vector polynomial $P$ of degree at most $m$ and $0<R_1<R_2$,
\begin{equation}
\sup_{B(R_2)}\abs P
\le (R_2/R_1)^m\sup_{B(R_1)}\abs P.
\label{eq:unif:polynomial-growth}
\end{equation}
To verify this, restrict each scalar dual pairing of $P$ to a
complex line through zero. The maximum principle in the exterior
of the radius-$R_1$ disc, applied to the polynomial divided by
$z^m$ including its removable value at infinity, gives the scalar
bound. Taking the supremum over unit dual vectors and lines proves
\eqref{eq:unif:polynomial-growth}.
\begin{proof}[Proof of Theorem~\ref{thm:uniformization}]
The preceding analytic and chart constructions, followed by
Propositions~\ref{prop:polynomial-models} and
\ref{prop:global-coordinates}, give a biholomorphism $\Psi:M\to\Omega$
onto an open set containing zero, with the forward smallness and
inverse-ball inclusion proved above. It remains to show that
$\Omega=\C^n$.
Suppose
$R_*=\sup\{R>0:B(R)\subset\Omega\}<\infty$.
It is positive because $0\in\Omega$ and $\Omega$ is open.
By \eqref{eq:dyn:forward-degree} there is an unbounded subsequence
with $\deg G_{0,k}\le Ds_k$ for some finite $D$.
Choose
$0<R_1<R_*<R_2$ so close to $R_*$ that
\begin{equation}
D\log(R_2/R_1)<\sigma/2-\gamma'.
\label{eq:unif:radii-choice}
\end{equation}
The closed smaller ball lies compactly in $\Omega$: it lies inside
some still larger centered ball of radius below $R_*$.
By \eqref{eq:unif:forward-smallness},
\eqref{eq:unif:polynomial-growth}, and
\eqref{eq:unif:radii-choice}, the chosen subsequence satisfies
\[
\sup_{B(R_2)}\abs{G_{0,k}}
\le \exp\left[
-\frac{\sigma}{2}s_k+Ds_k\log(R_2/R_1)\right]
< e^{-\gamma's_k}
\]
for all sufficiently large indices.
Equation~\eqref{eq:unif:inverse-ball-inclusion} then implies
$B(R_2)\subset\Omega$, contradicting the definition of $R_*$.
Therefore $R_*=\infty$, and $\Omega=\C^n$.
The map $\Psi$ in \eqref{eq:unif:global-map} is consequently a
biholomorphism $M\to\C^n$.
\end{proof}
The surjectivity step used the degrees
of the forward compositions $G_{0,k}$ on their available
subsequence; no degree estimate for their inverses, or uniform
instantaneous contraction ratio, is required.
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\end{document}