Hermitian pluriharmonic maps between almost Hermitian manifolds

TL;DR

This paper introduces Hermitian pluriharmonic maps between almost Hermitian manifolds, proving their relation to holomorphic maps and establishing energy monotonicity formulas with applications to holomorphicity results.

Contribution

It defines Hermitian pluriharmonic maps in the almost Hermitian setting and links them to holomorphic maps, providing new energy estimates and holomorphicity criteria.

Findings

Holomorphic and anti-holomorphic maps are Hermitian pluriharmonic.

Established monotonicity formulas for partial energies.

Derived holomorphicity results under energy growth conditions.

Abstract

In the case where both the domain and target manifolds are almost Hermitian, we introduce the concept of Hermitian pluriharmonic maps. We prove that any holomorphic or anti-holomorphic map between almost Hermitian manifolds is Hermitian pluriharmonic. We also establish some monotonicity formulae for the partial energies of Hermitian pluriharmonic maps into K\"ahler manifolds. As an application, under appropriate assumptions on the growth of the partial energies, some holomorphicity results are proven.