p-Laplace Operator and Diameter of Manifolds

Abstract

Let ${(M^{n},g)}$ be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the p-Laplacian ${\lambda_{1,p}(M)}$ and we prove that the limit of $p{\sqrt {\lambda_{1,p}(M)}}$ when $p\rightarrow\infty$ is 2/d(M), where d(M) is the diameter of M. Moreover, if ${(M^{n},g)}$ is an oriented compact hypersurface of the Euclidean space ${\mathbb{R}^{n+1}}$ or ${\mathbb{S}^{n+1}}$, we prove an upper bound of ${\lambda_{1,p}(M)}$ in terms of the largest principal curvature κ over M. As applications of these results, we obtain optimal lower bounds of d(M) in terms of the curvature. In particular, we prove that if M is a hypersurface of ${\mathbb{R}^{n+1}}$ then: $d(M)\ge\pi/\kappa$.