On the Calabi estimate of geometric flows of Hermitian metrics

TL;DR

This paper proves a general $C^1$ bound for Hermitian metric flows and applies it to improve regularity results for Hermitian curvature flows, including the second Chern-Ricci flow.

Contribution

It introduces a new $C^1$ a priori estimate and applies it to enhance regularity results for Hermitian curvature flows.

Findings

Established a $C^1$ a priori bound for Hermitian metric flows.

Obtained new regularity results for the second Chern-Ricci flow.

Improved understanding of geometric flow regularity in Hermitian geometry.

Abstract

We establish a general result ensuring a $C^1$ a priori bound for smooth curves of Hermitian metrics. As a main application, we obtain a new regularity result for Hermitian curvature flows, and in particular for the second Chern-Ricci flow.