Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow

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 $AdGL(T^{1,0}M)$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F:End(T^{1,0}M)\to \mathbb{R}$ 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 $\tilde{M}$.

20 pages, 1 figure

References in corpus (1)

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

Cited by in corpus (1)

• Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups