Liouville properties of plurisubharmonic functions

TL;DR

This paper establishes Liouville theorems for plurisubharmonic functions on complete Kähler manifolds, leading to splitting and gap theorems that deepen understanding of the geometric structure of these manifolds.

Contribution

It proves a Liouville theorem for plurisubharmonic functions and derives new splitting and gap theorems for complete Kähler manifolds with nonnegative bisectional curvature.

Findings

Liouville theorem for plurisubharmonic functions on complete Kähler manifolds

Splitting theorem for manifolds with nonnegative bisectional curvature

Optimal gap theorem for such manifolds

Abstract

We prove a Liouville theorem for the plurisubharmonic functions on complete Kaelher manifolds. As the applications, we prove a splitting theorem for complete Kaehler manifolds with nonnegative biscetional curvature in terms of the linear growth harmonic functions and a optomal gap theorem for such manifolds.