The Hermitian curvature flow on manifolds with non-negative Griffiths curvature

TL;DR

This paper investigates the Hermitian curvature flow on compact complex Hermitian manifolds, demonstrating that Griffiths positivity of the Chern curvature is preserved and that the flow exhibits regularization properties.

Contribution

It establishes the preservation of Griffiths positivity under the Hermitian curvature flow and analyzes the flow's regularization effects on the curvature's zero set.

Findings

Griffiths positive (non-negative) Chern curvature is preserved along the flow.

The zero set of the curvature becomes invariant under torsion-twisted parallel transport for t>0.

The flow exhibits regularization properties improving the curvature's behavior.

Abstract

In this paper we study a particular version of the Hermitian curvature flow (HCF) over a compact complex Hermitian manifold $(M,g,J)$. We prove that if the initial metric has Griffiths positive (non-negative) Chern curvature $\Omega$, then this property is preserved along the flow. On a manifold with Griffiths non-negative Chern curvature the HCF has nice regularization properties, in particular, for any $t>0$ the zero set of $\Omega(\xi,\bar\xi,\eta,\bar\eta)$ becomes invariant under certain torsion-twisted parallel transport.