Sharp estimate on the first eigenvalue of the p-Laplacian on compact manifold with nonnegative Ricci curvature

TL;DR

This paper establishes a precise estimate for the first nonzero eigenvalue of the p-Laplacian on compact manifolds with nonnegative Ricci curvature, using gradient comparison techniques and characterizing equality cases.

Contribution

It provides the sharp eigenvalue estimate for the p-Laplacian on such manifolds, extending previous results to include boundary cases with Neumann conditions.

Findings

Sharp eigenvalue estimate proven

Gradient comparison theorem developed

Equality cases characterized

Abstract

We prove the sharp estimate on the first nonzero eigenvalue of the p-laplacian on a compact Riemannian manifold with nonnegative Ricci curvature and possibly with convex boundary (in this case we assume Neumann b.c. on the p-laplacian). The proof is based on a gradient comparison theorem. We will also charachterize the equality case in the estimate.