Bounded holomorphic functions on negatively curved K\"ahler manifolds of dimension $\ge 3$

TL;DR

This paper proves the existence of non-constant bounded holomorphic functions on certain negatively curved Kähler manifolds of dimension three or higher, using advanced complex analysis techniques.

Contribution

It establishes the existence of bounded holomorphic functions on negatively curved Kähler manifolds of dimension ≥3, extending previous results in complex geometry.

Findings

Existence of non-constant bounded holomorphic functions on M

Use of bounded plurisubharmonic exhaustion functions

Application of CR functions and Holder estimates

Abstract

Let M be a simply-connected complete Kahler manifold whose sectional curvature is bounded between two negative numbers. In this paper we prove the existence of non-constant bounded holomorphic functions on M if the complex dimension of M is greater or equal to three. Our proof uses bounded plurisubharmonic exhaustion functions, the Cauchy-Riemann equations and uniform Holder estimates for CR functions on geodesic spheres.