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

摘要

The aim of this paper is give a simple proof of some results in \cite{JunLing-2006-IJM} and \cite{JunLing-2007-AGAG}, which are very deep studies in thesharp lower bound of the first eigenvalue in the Laplacian operator on compactRiemannian manifolds with nonnegative Ricci curvature. We also get a resultabout 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 methodsused in here due to \cite{Jun Ling-2006-IJM} (or \cite{JunLing-2007-AGAG}) onthe whole, to some extent we can tackle the singularity of test functions andalso simplify greatly much calculation in these references. Maybe this providesanother way to estimate eigenvalues.