The Drift Laplacian and Hermitian Geometry

TL;DR

This paper establishes a lower bound for the lowest eigenvalue of the complex Laplacian on compact Hermitian manifolds, linking it to geometric properties and the deviation from Kähler structure.

Contribution

It introduces a new eigenvalue estimate for the complex Laplacian on Hermitian manifolds, connecting spectral bounds with geometric and curvature conditions.

Findings

Lower bound for the complex Laplacian eigenvalue depending on geometric data.

Estimation of principal eigenvalue of a drift Laplacian.

Structural insights into Hermitian manifolds using recent results.

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 $\lambda \geq C$ 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 K\"ahler. 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.