A Chern-Calabi flow on Hermitian manifolds

TL;DR

This paper investigates a flow analogous to the Calabi flow on compact Hermitian manifolds, establishing conditions for convergence to a unique Chern-Ricci-flat metric under bounded curvature assumptions.

Contribution

It introduces a Chern-Calabi flow in the non-Kähler setting and proves convergence to a Chern-Ricci-flat metric given uniform scalar curvature bounds.

Findings

Established a priori estimates for the flow.

Proved convergence to a Chern-Ricci-flat metric under bounded scalar curvature.

Identified conditions for smooth convergence in the Hermitian setting.

Abstract

We study an analogue of the Calabi flow in the non-K\"ahler setting for compact Hermitian manifolds with vanishing first Bott-Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern-Ricci-flat metric in the $\partial\bar{\partial}$-class of the initial metric.