The Drift Laplacian and Hermitian Geometry

Abstract

Let $(M^n,h)$ be a compact Hermitian manifold. Suppose $\lambda$ is the lowest eigenvalue of the complex Laplacian on $M$. We prove that $\lambda \ge C$ where $C$ depends only on the dimension $n$, the diameter $d$, the Ricci curvature of the Levi-Civita connection on $M$, and a norm, expressed in curvature, that determines how much $M$ fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng. We combine these results to obtain the main estimate.

28 pages. This version replaces the last one. We have updated some of the results and proved Conjecture 5 in some special cases. We plan to post a second part soon and wanted a more current version online

References in corpus (1)

• On Curvature Tensors of Hermitian Manifolds