On Eigenvalue Problem Of Laplace Operator On Riemann Manifold

In this paper,we mainly study the eigenvalue estimation of Laplace operators on Riemann manifold,that is,the lower bound of the first eigenvalue of Laplace operator and the lower bound of higher order eigenvalues under different conditions of Ricci curvature.We reviewe the Pólya conjecture of Dirichlet eigenvalues and some eigenvalue problems under some new operators in this paper.Firstly,we study the lower bound estimation of the first eigenvalue of the Laplace operator.By means of the maximum principle and the method of gradient estimation,we obtain the lower bound estimates of the first no boundary or Neumann eigenvalue and Dirichlet eigenvalue on Riemann manifold.Secondly,we study the lower bound of higher eigenvalue under different Ricci curvature conditions.Specifically,there are two curvature assumptions of Ricci curvature with nonnegative lower bound and negative lower bound.Especially the curvature condition of Ricci curvature with negative lower bound,we obtained the result which is a quantitative lower bound estimate rather than a qualitative one in this paper.Thirdly,we review the Pólya conjecture of Dirichlet eigenvalues.We mainly review the proof methods of He and Liu in improving the original literature,and points out the shortcomings of the method of He and Liu.Finally,we review the result of the eigenvalue problom of p-Laplace operator and Laplace operator eigenvalue problem under Ricci flow.And we give a supplementary proof of the lower bound estimation of the first eigenvalue of p-Laplace operators in the K?hler manifold with integral curvature,which has not been proved in the original literature.