Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

TL;DR

This paper establishes sharp lower bounds for the first nonzero eigenvalue of the p-Laplacian on compact manifolds with convex boundary and negative Ricci curvature, using advanced gradient comparison methods.

Contribution

It provides the first sharp eigenvalue estimates for the p-Laplacian under negative Ricci curvature bounds with Neumann boundary conditions.

Findings

Sharp lower bounds for the first eigenvalue of the p-Laplacian.

Extension of eigenvalue estimates to manifolds with negative Ricci curvature.

Use of refined gradient comparison techniques.

Abstract

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $\Delta_p$ when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.