Lower bounds for the first eigenvalue of $p$-Laplacian on Kähler manifolds

Abstract

We study the eigenvalue problem for the $p$-Laplacian on Kähler manifolds. Our first result is a lower bound for the first nonzero eigenvalue of the $p$-Laplacian on compact Kähler 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 Kähler manifolds with smooth boundary for $p\in (1,\infty )$. Our results generalize corresponding results for the Laplace eigenvalues on Kähler manifolds proved in [14].

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