Lower bounds for the first eigenvalue of the Laplacian on K\"ahler manifolds

TL;DR

This paper derives lower bounds for the first nonzero eigenvalue of the Laplacian on closed and bounded K"ahler manifolds, extending classical Riemannian results to the K"ahler setting using geometric curvature bounds.

Contribution

It provides new lower bounds for Laplacian eigenvalues on K"ahler manifolds based on curvature, dimension, and diameter, generalizing known Riemannian eigenvalue estimates.

Findings

Lower bounds for eigenvalues on closed K"ahler manifolds.

Lower bounds for Neumann and Dirichlet eigenvalues on manifolds with boundary.

Extension of Riemannian eigenvalue bounds to K"ahler geometry.

Abstract

We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed K\"ahler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact K\"ahler manifolds with boundary, we prove lower bounds for the first nonzero Neumann or Dirichlet eigenvalue in terms of geometric data. Our results are K\"ahler analogues of well-known results for Riemannian manifolds.