Hessian comparison and eigenvalue estimate of almost Hermitian manifolds

TL;DR

This paper extends Hessian and eigenvalue estimates to almost Hermitian manifolds using Bochner techniques, generalizing previous Laplacian comparisons and providing sharp bounds for specific subclasses.

Contribution

It introduces new Hessian comparison theorems and eigenvalue estimates for almost Hermitian, quasi Kähler, and nearly Kähler manifolds, broadening geometric analysis tools.

Findings

Derived a complex Hessian comparison for almost Hermitian manifolds.

Reproved a diameter estimate for almost Hermitian manifolds.

Established sharp eigenvalue bounds on quasi Kähler and nearly Kähler manifolds.

Abstract

In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain a sharp eigenvalue estimate on quasi Kaehler manifolds and a sharp Hessian comparison on nearly Kaehler manifolds.