Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow

TL;DR

This paper demonstrates that the Hermitian Curvature Flow preserves various natural curvature positivity conditions on Hermitian manifolds, and explores implications for the existence of periodic solutions related to the manifold's canonical bundle.

Contribution

It introduces a framework for curvature conditions preserved by HCF using invariant convex sets, extending understanding of curvature preservation under geometric flows.

Findings

HCF preserves Griffiths positivity and Dual-Nakano positivity.

HCF maintains positivity of holomorphic orthogonal bisectional curvature.

Periodic solutions to HCF imply the manifold's universal cover has trivial canonical bundle.

Abstract

The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we construct a convex set of curvature operators $C(S,F)$, which is invariant under the HCF. Varying $S$ and $F$, we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature. As an application, we prove that periodic solutions to the HCF can exist only on manifolds $M$ with the trivial canonical bundle on the universal cover $\widetilde{M}$.