Holomorphic functions on certain K\"{a}hler manifolds

TL;DR

This paper studies Liouville theorems and dimension bounds for holomorphic functions with exponential growth on complete Kähler manifolds, independent of curvature conditions, motivated by Ricci flow applications.

Contribution

It establishes general Liouville theorems and dimension estimates for exponential holomorphic functions on Kähler manifolds without curvature assumptions.

Findings

Liouville theorems for exponential holomorphic functions

Dimension estimates for function spaces

Results applicable to Ricci solitons

Abstract

We investigate Liouville theorems and dimension estimates for the space of exponentially growing holomorphic functions on complete K\"{a}hler manifolds. While our work is motivated by the study of gradient Ricci solitons in the theory of Ricci flow, the most general results we prove here do not require any knowledge of curvature.