On holomorphic functions on negatively curved manifolds

TL;DR

This paper explores the properties of holomorphic functions on negatively curved Kähler manifolds, showing how their real parts and moduli are contractive with respect to hyperbolic distances, extending classical results to curved settings.

Contribution

It extends Yau's theorem to show contractivity of holomorphic functions' real parts and moduli on negatively curved Kähler manifolds, linking geometric curvature bounds to function behavior.

Findings

Real part of holomorphic functions is contractive with respect to hyperbolic distance.

Modulus of bounded holomorphic functions is contractive on the manifold.

Results generalize classical hyperbolic function properties to negatively curved manifolds.

Abstract

Based on a well known Sh.-T. Yau theorem we obtain that the real part of a holomorphic function on a K\"{a}hler manifold with the Ricci curvature bounded from below by $-1$ is contractive with respect to the distance on the manifold and the hyperbolic distance on $(-1,1)$ inhered from the domain $(-1,1)\times\mathbb{R}$. Moreover, in the case of bounded holomorphic functions we prove that the modulus is contractive with respect to the distance on the manifold and the hyperbolic distance on the unit disk.