Existence and geometry of Hermitian metrics with constant second scalar curvature

TL;DR

This paper investigates the existence and geometric properties of Hermitian metrics with constant second scalar curvature on compact manifolds, exploring related elliptic equations and conditions leading to Kähler-Einstein metrics.

Contribution

It introduces a Yamabe-type problem for the second Bismut scalar curvature and analyzes elliptic equations in Hermitian conformal classes, linking curvature conditions to Kähler-Einstein metrics.

Findings

Solved a Yamabe-type problem for second Bismut scalar curvature.

Derived geometric consequences from elliptic equations in Hermitian conformal classes.

Established conditions under which Hermitian metrics imply Kähler-Einstein metrics.

Abstract

We study Hermitian metrics with constant second scalar curvature on compact manifolds. We first consider a Yamabe-type problem for the second Bismut scalar curvature under a natural topological condition, and then analyze elliptic equations arising from constant second Chern scalar curvature within a fixed Hermitian conformal class and derive geometric consequences. Finally, under an Einstein-type condition on the second Chern curvature, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, which in certain cases further implies the existence of a K\"ahler-Einstein metric.