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

Abstract

We study the first nonzero eigenvalues for the $p$-Laplacian on quaternionic K\"ahler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the $p$-Laplacian on compact quaternionic K\"ahler manifolds. Our second result is a lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on compact quaternionic K\"ahler manifolds with smooth boundary. Our results generalize corresponding results for the Laplacian eigenvalues on quaternionic K\"ahler manifolds proved in [22].