Deformation of Hermitian metrics

TL;DR

This paper investigates how Hermitian metrics with specific curvature properties can be deformed to achieve positive or negative curvature, extending classical methods to the Hermitian setting.

Contribution

It adapts the conformal perturbation method to Hermitian metrics, enabling deformation of metrics with quasi-positive or quasi-negative second Chern-Ricci curvature.

Findings

Hermitian metrics with quasi-positive second Chern-Ricci curvature can be deformed to positive curvature

Hermitian metrics with quasi-negative second Chern-Ricci curvature can be deformed to negative curvature

Extension of classical conformal perturbation methods to Hermitian geometry

Abstract

In this work, we study the deformation of Hermitian metrics with Chern connection. By adapting the conformal perturbation method of Aubin and Ehrlich to Hermitian setting, we prove that Hermitian metrics with quasi-positive (resp. quasi-negative) second Chern-Ricci curvature can be deformed to one with positive (resp. negative) curvature.