Lie-algebraic curvature conditions preserved by the Hermitian curvature flow

2017/10/17 by Yury Ustinovskiy
Mathematics · #Algebraic Geometry and Number Theory #Curvature #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Hermitian manifold #Hermitian matrix #Holomorphic function #Invariant (physics) #Mean curvature flow #Regular polygon #Scalar curvature #math.CV #math.DG

paper · pdf · doi:10.1007/s00208-020-01965-y

20 pages, 1 figure

arxiv created 2017/10/17 · openalex created_date 2017/11/10 · openalex publication_date 2020/03/04 · arxiv updated 2020/09/03 · openalex updated_date 2026/08/05

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^1,0M)-invariant subset S⊂ End(T^1,0M) and a ncie function F\colon End(T^1,0M)→\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 \widetildeM.

Citations

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• Four-manifolds with positive curvature operator
• Classification of manifolds with weakly 1/4-pinched curvatures
• La 1-forme de torsion d'une vari�t� hermitienne compacte
• Manifolds with positive curvature operators are space forms
• Manifolds with positive curvature
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• Manifolds with 1/4-pinched curvature are space forms