Hessian Comparison and Spectrum Lower Bound of Almost Hermitian Manifolds

Abstract

In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain a sharp eigenvalue estimate on quasi Kähler manifolds and a sharp Hessian comparison on nearly Kähler manifolds. 2000 Mathematics Subject Classification. Primary 53B25; Secondary 53C40.

Schwarz lemma: the case of equality and an extension

“…It is smooth away from o and its cut locus. The following complex Hessian comparison theorem on almost Hermitian manifolds due to Tosatti [19] will be used in the proof of Theorem 1.6 (also see [25]).…”