First eigenvalue estimates on complete K\"ahler manifolds

TL;DR

This paper establishes a lower bound for the first eigenvalue of the Laplacian on complete K"ahler manifolds with holomorphic sectional curvature at least 2, using a new Bochner-Kodaira identity.

Contribution

It introduces a novel Bochner-Kodaira type identity tailored for holomorphic sectional curvature to derive eigenvalue estimates.

Findings

First eigenvalue bound depends on complex dimension n.

Lower bound explicitly expressed in terms of n.

Method applicable to complete K"ahler manifolds with curvature constraints.

Abstract

Let $ (M,\omega_g) $ be a complete K\"ahler manifold of complex dimension $n$. We prove that if the holomorphic sectional curvature satisfies $\mathrm{HSC} \geq 2 $, then the first eigenvalue $\lambda_1$ of the Laplacian on $(M,\omega_g)$ satisfies $$ \lambda_1 \geq \frac{320(n-1)+576}{81(n-1)+144}.$$ This result is established through a new Bochner-Kodaira type identity specifically developed for holomorphic sectional curvature.