On the Gauduchon Curvature of Hermitian Manifolds

TL;DR

This paper explores the properties of Gauduchon curvature on Hermitian manifolds, establishing new existence results for Ricci-flat metrics, analyzing curvature monotonicity, and revealing rigidity phenomena related to holomorphic sectional curvatures.

Contribution

It extends known curvature properties to all Gauduchon connections, proves existence of Ricci-flat metrics on certain manifolds, and uncovers curvature rigidity and maximality results.

Findings

Existence of t-Gauduchon Ricci-flat metrics on suspensions of Sasaki–Einstein manifolds.

Monotonicity of Gauduchon holomorphic sectional curvature.

Rigidity when Hermitian metrics have equal Gauduchon holomorphic sectional curvatures.

Abstract

It is shown that many results, previously believed to be properties of the Lichnerowicz Ricci curvature, hold for the Ricci curvature of all Gauduchon connections. We prove the existence of $t$--Gauduchon Ricci-flat metrics on the suspension of a compact Sasaki--Einstein manifold, for all $t \in (-\infty,1)$; in particular, for the Bismut, Minimal, and Hermitian conformal connection. A monotonicity theorem is obtained for the Gauduchon holomorphic sectional curvature, illustrating a maximality property for the Chern connection and furnishing insight into known phenomena concerning hyperbolicity and the existence of rational curves. Moreover, we show a rigidity result for Hermitian metrics which have a pair of Gauduchon holomorphic sectional curvatures that are equal, elucidating a duality implicit in the recent work of Chen--Nie.