Tamed exhaustion functions and Schwarz type lemmas for almost Hermitian manifolds

TL;DR

This paper develops exhaustion functions on almost Hermitian manifolds, leading to Schwarz lemmas and Liouville theorems for almost holomorphic maps, advancing understanding of geometric analysis in this setting.

Contribution

It introduces a new class of exhaustion functions and derives Schwarz lemmas and Liouville theorems for almost Hermitian manifolds, extending classical results to this broader context.

Findings

Existence of special exhaustion functions on almost Hermitian manifolds

Schwarz type lemmas for almost holomorphic maps

Liouville theorems for bounded almost holomorphic maps

Abstract

In this paper, we study a special exhaustion function on almost Hermitian manifolds and establish the existence result by using the Hessian comparison theorem. From the viewpoint of the exhaustion function, we establish related Schwarz type lemmas for almost holomorphic maps between two almost Hermitian manifolds. As corollaries, we deduce Liouville type theorems for almost holomorphic maps.