First eigenvalue estimates on complete balanced Hermitian manifolds

TL;DR

This paper extends classical Riemannian eigenvalue estimates to complete balanced Hermitian manifolds, relating the first eigenvalue of the Laplace-de Rham operator to holomorphic curvature measures.

Contribution

It provides new eigenvalue estimates for Hermitian manifolds using holomorphic Ricci and sectional curvatures, generalizing Riemannian results.

Findings

Eigenvalue bounds in terms of holomorphic Ricci curvature

Eigenvalue bounds in terms of holomorphic sectional curvature

Extension of classical estimates to Hermitian geometry

Abstract

In analogy with classical results in Riemannian geometry, we establish estimates for the first eigenvalue of the Laplace-de Rham operator on complete balanced Hermitian manifolds in terms of either the holomorphic Ricci curvature or the holomorphic sectional curvature associated with the Strominger-Bismut connection.