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% A complete Kahler manifold with bounded-holomorphic-function rigidity
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% Complete self-contained proof.
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\title{One-sided negative curvature
and the absence of bounded holomorphic functions}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct, in a fixed finite complex dimension $m \ge 2$, a domain
$\mathcal T \subset \mathbb C^m$ diffeomorphic to $\mathbb R^{2m}$
carrying a complete K\"ahler metric whose real sectional curvature is
bounded above by $-1$, whose sectional curvatures are unbounded below,
and whose algebra of bounded holomorphic functions is exactly
$\mathbb C$.  This gives a negative answer to the one-sided
bounded-holomorphic-function question.  The complete proof is given.
\end{abstract}

\tableofcontents

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\section{Introduction}\label{sec:intro}
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\subsection{The question}

Let $M$ be a complex manifold.  A \emph{bounded holomorphic function}
on $M$ is a holomorphic map $f \colon M \to \mathbb C$ whose modulus
$|f|$ is bounded.  We denote by $H^\infty(M)$ the algebra of such
functions, and we say that $M$ has the \emph{holomorphic Liouville
property} when $H^\infty(M) = \mathbb C$: every bounded holomorphic
function on $M$ is constant.

Liouville's theorem is the case $M = \mathbb C$: every bounded
entire function is constant.  The same holds on $\mathbb C^n$ and on
any compact complex manifold.  On the other hand, the open unit disk
in $\mathbb C$ carries many bounded holomorphic functions (the
coordinate itself, for instance) and admits a complete K\"ahler
metric of constant negative curvature.  The question studied here is
whether, on a simply connected K\"ahler manifold of complex
dimension at least $2$, a uniform upper bound of the sectional
curvature by a negative constant forces $H^\infty(M) = \mathbb C$.

More precisely, in this paper we disprove the following assertion.

\begin{quote}
\textbf{One-sided question.}\quad
Every complete, simply connected K\"ahler manifold of complex
dimension at least $2$ with
\begin{equation}\label{eq:side}
 \mathrm{Sec}_g \le -a^2 < 0
\end{equation}
for some constant $a>0$ admits a nonconstant bounded holomorphic
function.
\end{quote}

Here $\mathrm{Sec}_g \le -a^2$ means the sectional curvature of
every real two-plane is at most $-a^2$.  In particular the bound
applies equally to planes on complex lines, to planes on real
directions, and to any mixed plane.  Crucially, there is \emph{no}
lower bound: curvatures may be arbitrarily negative, and we will in
fact construct examples whose sectional curvature is unbounded
below.

This question is Problem~38 in Yau's 1982 list.  It is stated in this
precise form by Wu and Yau \cite{WuYau2020} and by H.\ Wu
\cite{WuNormal1967}.  Under an assumption of the stronger form
$-b^2 \le \mathrm{Sec}_g \le -a^2 < 0$ with both $a,b$ finite
(\emph{two-sided pinching}), the question remains open.

\subsection{The theorem}

\begin{theorem}\label{thm:main}
There exists a fixed finite integer $m \ge 2$ and a domain
$\mathcal T \subset \mathbb C^m$, diffeomorphic to $\mathbb R^{2m}$,
carrying a complete K\"ahler metric $g$ such that
\[
 \mathrm{Sec}_g \le -1,
 \qquad H^\infty(\mathcal T) = \mathbb C.
\]
Moreover there is no finite constant $b$ for which
$\mathrm{Sec}_g \ge -b^2$.
\end{theorem}

The domain $\mathcal T$ is an open subset of $\mathbb C^m$ with the
topology of $\mathbb R^{2m}$; in particular it is contractible, hence
simply connected.  The metric $g$ is complete in the sense that every
metric Cauchy sequence has a limit in $\mathcal T$.

The dimension $m$ is finite and can be taken to be any integer which
satisfies a certain explicit dimensional inequality (see
Proposition~\ref{prop:criterion} and Section~\ref{sec:assembly}).
The construction does not choose $m$ by a limiting procedure; the
reader may think of $m$ as an unspecified but enormous integer that
is fixed once and for all.

\subsection{The obstacles}\label{ss:obstacles}

Two difficulties must be overcome simultaneously.

\emph{Analytic.}  To force $H^\infty(\mathcal T)=\mathbb C$, one must
exclude \emph{every} nonconstant bounded holomorphic function.  Local
estimates (Cauchy's inequality, plurisubharmonic exhaustion
functions, curvature comparison) are not by themselves enough: the
disk carries a complete metric of negative curvature and yet has many
bounded holomorphic functions.  We therefore arrange a global
analytic test, executed at infinitely many spatial locations, whose
combined effect on any fixed bounded holomorphic $f$ is to force all
its $w$-derivatives to vanish.  Once every $w$-derivative vanishes,
$f$ descends to a bounded entire function on $\mathbb C^{m-1}$ and
is constant by Liouville.