On the evolution of a Hermitian metric by its Chern-Ricci form

TL;DR

This paper studies the evolution of Hermitian metrics on complex manifolds via their Chern-Ricci form, extending known flows and analyzing their behavior on various complex surfaces and manifolds.

Contribution

It introduces new results on the maximal existence time of the flow and explores its behavior on specific classes of complex manifolds, including Hopf and Gauduchon manifolds.

Findings

Maximal existence time characterized in terms of initial data

Flow behavior analyzed on complex surfaces and Hopf manifolds

New estimate for the complex Monge-Ampère equation on Hermitian manifolds

Abstract

We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kahler-Ricci flow if the initial metric is Kahler. We find the maximal existence time for the flow in terms of the initial data. We investigate the behavior of the flow on complex surfaces when the initial metric is Gauduchon, on complex manifolds with negative first Chern class, and on some Hopf manifolds. Finally, we discuss a new estimate for the complex Monge-Ampere equation on Hermitian manifolds.