==== TABLE OF CONTENTS ====
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//bibliography.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//main.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/charts.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/discs.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/dynamics.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/ellipsoids.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/flow.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/introduction.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/preliminaries.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/uniformization.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/variation.tex
Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//unicode-map.tex
===========================

==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//bibliography.tex ====
\begin{thebibliography}{99}

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==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//bibliography.tex ====

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\title{Uniformization of complete K\"ahler manifolds\protect\\
with positive bisectional curvature}
\author{OpenAI}
\date{September 23, 2026}
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\begin{document}
\maketitle
\begin{abstract}
We prove that every complete connected noncompact K\"ahler manifold with
strictly positive holomorphic bisectional curvature is biholomorphic to
complex Euclidean space. This resolves Yau's uniformization conjecture
positively in every complex dimension.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/flow}
\input{sections/discs}
\input{sections/variation}
\input{sections/ellipsoids}
\input{sections/charts}
\input{sections/dynamics}
\input{sections/uniformization}
\input{bibliography}
\end{document}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//main.tex ====

==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/charts.tex ====
\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(00$.
\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 00$
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
$\dist_{\widetilde g}(z,y)0$ 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 $a_01$ 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 $\delta0$.
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
$00$, 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 $R0$,
\[
 \log\sup_{B(e^{-\gamma s})}|F_{s,u}|
 \le-\theta_s\sigma u+(1-\theta_s)\log B_*,
 \qquad
 \theta_s=\frac{\log(R/e^{-\gamma s})}
                 {\log(R/e^{-C_0s})}
 \ge\frac{\gamma}{2C_0}
\]
for all sufficiently large $s$.  In this formula choose
$\gamma_00$ and $\sigma>0$ such that
\begin{equation}\label{disc:extra-integrability}
 \int_{0<|z|0.
\]
These images extend across zero by the same $L^2$ argument.  The set
$S$ is an $R$-submodule: for a sum take the smaller of the two positive
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 vector subspace and is closed.  Jet convergence of
$Aa_m\in S$ consequently gives
\[
 Aa\in S+\mathfrak m^kR^q\qquad\hbox{for every }k.
\]
Krull's intersection Theorem for the finite local module $Q$ gives
$\bigcap_k\mathfrak m^kQ=0$ \cite[Lemma 10.51.4]{StacksKrull}.
Thus $Aa\in S$.  By definition 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 $a_m$ 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 has the same 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 $\Omega_j$.
Assume that for each compact $K\subset X$, the functions $u_j$ are
uniformly bounded above on $K$ once $K\subset\Omega_j$.
Every sequence has a subsequence for which either $u_j$ tends 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 moving subsets of a fixed compact set of uniformly
positive measure suffice.  If the $L^1_{\mathrm{loc}}$ limit is the
constant $c$, then
\[
 \limsup_j\sup_K u_j\le c
\]
for every compact $K\subset X$.
\end{lemma}

\begin{proof}
We explain the lower-anchor step rather than building it into the
compactness theorem.  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 functions are nonnegative and 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.

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 therefore give $L^1$ bounds on
every fixed compact subset of $X$.

Families of subharmonic functions bounded in $L^1_{\mathrm{loc}}$
are relatively compact in that topology \cite[Chapter~I, Proposition~4.21]{DemaillyBook}.
Apply this result 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 $-\infty$; this is the other alternative.  This also proves
the assertion for exhausting domains, since each finite chain of
balls is contained in every sufficiently late $\Omega_j$.

Finally 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}]
We first control centers of discs $f_l$ in $M$.  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, with a fixed normalization
constant.  In particular, the functions
\[
 E_l(\theta)=\int_{1/2}^1
          |\partial_rf_l(re^{i\theta})|_g^2r\,dr
\]
have uniformly bounded angular integrals.  On a set of angles of
measure bounded below, $E_l(\theta)\le C(A_0+1)$.  For each such
angle and every $r\in[1/2,3/4]$, Cauchy--Schwarz gives
\[
 \dist_g(f_l(re^{i\theta}),K)
 \le E_l(\theta)^{1/2}
       \left(\int_{1/2}^1\frac{dr}{r}\right)^{1/2}.
\]
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 function
$\psi_\infty$; 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.

We spell out the required 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)0.
\]
For small enough $r$, $F\circ f$ on $0<|z|0$ 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}
==== END Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/discs.tex ====

==== BEGIN Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature//sections/dynamics.tex ====
\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
$00,\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 $00$ 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