Sharp estimate of lower bound for the first eigenvalue in the Laplacian operator on compact Riemannian manifolds

TL;DR

This paper simplifies the proof of sharp lower bounds for the first eigenvalue of the Laplacian on compact Riemannian manifolds with nonnegative Ricci curvature, also providing an optimal estimate for the Neumann case.

Contribution

It offers a simpler proof of existing deep results and extends to an optimal lower bound for the Neumann eigenvalue in specific cases.

Findings

Simplified proof of sharp lower bounds for the first eigenvalue.

Extended results to the Neumann eigenvalue with optimal bounds.

Provided an alternative approach to eigenvalue estimation.

Abstract

The aim of this paper is give a simple proof of some results in \cite{Jun Ling-2006-IJM} and \cite{JunLing-2007-AGAG}, which are very deep studies in the sharp lower bound of the first eigenvalue in the Laplacian operator on compact Riemannian manifolds with nonnegative Ricci curvature. We also get a result about lower bound of the first Neumann eigenvalue in a special case. Indeed, our estimate of lower bound in the this case is optimal. Although the methods used in here due to \cite{Jun Ling-2006-IJM} (or \cite{JunLing-2007-AGAG}) on the whole, to some extent we can tackle the singularity of test functions and also simplify greatly much calculation in these references. Maybe this provides another way to estimate eigenvalues.