Estimate Of The First Eigenvalue On Compact Manifolds

In this paper,we use the method of the estimated characteristic function to estimate a lower bound of the first eigenvalue on compact Riemann manifold.This paper is made up of five parts:In chapter one,we introduce the background and investigation meaning of esti-mate of the first eigenvalues.Furthermore,we show the relevant research progress.In chapter two,we give the basic concepts and basic formulas about the first eigenvalues.In chapter three,we'll describe the classification of the problem and the results to be certified.In chapter four,we give the proof of some results about the lower bound for the first eigenvalues with non-boundary condition.In chapter five,we give the proof of some results about the lower bound for the first eigenvalues with three different boundary conditions.