Hermitian Curvature Flow

TL;DR

This paper introduces a new curvature-based functional for Hermitian metrics, derives an associated elliptic equation, and studies a related parabolic flow, establishing existence, regularity, and stability results near K"ahler-Einstein metrics.

Contribution

It defines a novel Hermitian curvature functional, formulates a related flow, and proves foundational results including existence, regularity, and stability near special metrics.

Findings

Short time existence and regularity of the flow

Stability of the flow near K"ahler-Einstein metrics with negative or zero first Chern class

Relation of solutions to K"ahler-Einstein metrics

Abstract

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to K\"ahler-Einstein metrics, and are automatically K\"ahler-Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existence and regularity results for this flow, as well as stability for the flow near K\"ahler-Einstein metrics with negative or zero first Chern class.