On pseudoholomorphic map between almost Hermitian manifolds

TL;DR

This paper investigates pseudoholomorphic maps between almost Hermitian manifolds using the canonical connection, deriving estimates, Liouville theorems, and inequalities to deepen understanding of their geometric properties.

Contribution

It introduces the use of the canonical connection instead of Levi-Civita for studying pseudoholomorphic maps, providing new estimates and theorems in this context.

Findings

Established $C^2$-estimate of canonical second fundamental form

Proved Liouville type theorems for pseudoholomorphic maps

Derived Simons integral inequality for pseudoholomorphic isometric immersions

Abstract

In this paper, we use the canonical connection instead of Levi-Civita connection to study the smooth maps between almost Hermitian manifolds, especially, the pseudoholomorphic ones. By using the Bochner formulas, we obtian the $C^2$-estimate of canonical second fundamental form, Liouville type theorems of pseudoholomorphic maps, pseudoholomorphicity of pluriharmonic maps, and Simons integral inequality of pseudoholomorphic isometric immersion.