On the fundamental tone of the $p$-Laplacian on Riemannian manifolds and applications

TL;DR

This paper establishes a general lower bound for the fundamental tone of the p-Laplacian on Riemannian manifolds with specific functions, with applications to negatively curved, warped product, and Riemannian submersion manifolds.

Contribution

It provides a new lower bound for the p-Laplacian's fundamental tone applicable to various classes of Riemannian manifolds, extending previous results.

Findings

Derived a general lower bound for the p-Laplacian's fundamental tone.

Applied the bound to negatively curved simply connected manifolds.

Extended results to warped product manifolds and Riemannian submersions.

Abstract

We present a general lower bound for the fundamental tone for the $p$-Laplacian on Riemannian manifolds carrying a special kind of function. We then apply our result to the cases of negatively curved simply connected manifolds, a class of warped product manifolds and for a class of Riemannian submersions.