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

Abstract

We study first nonzero eigenvalues for the p p -Laplacian on Kähler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the p p -Laplacian on compact Kähler manifolds in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature for p ∈ ( 1 , 2 ] p\in (1, 2] . Our second result is a sharp lower bound for the first Dirichlet eigenvalue of the p p -Laplacian on compact Kähler manifolds with smooth boundary for p ∈ ( 1 , ∞ ) p\in (1, \infty ) . Our results generalize corresponding results for the Laplace eigenvalues on Kähler manifolds proved by Li and Wang [Trans. Amer. Math. Soc. 374 (2021), pp. 8081–8099].