The modified Calabi-Yau problems for CR-manifolds and applications

TL;DR

This paper investigates the existence of bounded holomorphic functions on negatively curved Kähler manifolds, providing explicit bounded contact forms and structures at infinity, and discusses related open problems in CR and Alexandrov spaces.

Contribution

It establishes the existence of explicit bounded contact forms and structures at infinity for certain negatively curved Kähler manifolds, addressing a question of Yau.

Findings

Existence of explicit bounded contact form on the manifold

Sphere at infinity admits a bounded contact structure

Discussion of open problems in Calabi-Yau and Yau's questions

Abstract

In this paper, we derive a partial result related to a question of Yau: "Does a simply-connected complete K\"ahler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?" Main Theorem. Let $M^{2n}$ be a simply-connected complete K\"ahler manifold M with negative sectional curvature $ \le -1 $ and $S_\infty(M)$ be the sphere at infinity of $M$. Then there is an explicit {\it bounded} contact form $\beta$ defined on the entire manifold $M^{2n}$. Consequently, the sphere $S_\infty(M)$ at infinity of M admits a {\it bounded} contact structure and a bounded pseudo-Hermitian metric in the sense of Tanaka-Webster. We also discuss several open modified problems of Calabi and Yau for Alexandrov spaces and CR-manifolds.