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

TL;DR

This paper establishes new lower bounds for the first eigenvalue of the $p$-Laplacian on compact K"ahler manifolds, extending known results for Laplace eigenvalues to a broader class of nonlinear operators.

Contribution

It provides the first lower bounds for the $p$-Laplacian eigenvalues on K"ahler manifolds, incorporating geometric curvature bounds and generalizing previous Laplace eigenvalue results.

Findings

Lower bound for the first nonzero eigenvalue in terms of geometric quantities

Sharp lower bound for the first Dirichlet eigenvalue with boundary

Generalization of classical Laplace eigenvalue bounds to $p$-Laplacian

Abstract

We study the eigenvalue problem for the $p$-Laplacian on K\"ahler manifolds. Our first result is a lower bound for the first nonzero eigenvalue of the $p$-Laplacian on compact K\"ahler manifolds in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature for $p\in (1, 2]$. Our second result is a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on compact K\"ahler manifolds with smooth boundary for $p\in (1, \infty)$. Our results generalize corresponding results for the Laplace eigenvalues on K\"ahler manifolds proved in [14].