Eigenvalue Comparisons And Some Eigenvalue Estimates

The study of Spectral properties of Laplacian on compact Riemannian manifolds(with or without boundary)and non-compact complete Riemannian manifolds is an im-portant topic in Riemannian geometry.The Steklov eigenvalue problem was proposed by Stekloff in 1902,which has significance in some fields of physics,such as fluid mechan-ics,electromagnetics,etc..The Laplacian subject to the Wentzell boundary conditions can be seen as a natural extension of the classical Steklov eigenvalue problem and attracts attention in rencent years.In this paper,we mainly study the eigenvalue comparison the-orems of these two eigenvalue problems.Meanwhile,some estimates for eigenvalues of different type will also be shown.More precisely,we have obtained the followings:1)For a given n-dimensional(n?2)complete Riemannian manifold(Mⁿ,g)having a radial sectional curvature upper bound with regard to the point p?M.For n=2,3,the first non-zero Steklov eigenvalue of geodesic ball within the cut locus of this point on these manifolds can be bounded from above by that of the geodesic ball with the same radius and the center p⁺in the spherically symmetric manifolds with the base point p⁺determined by the curvature bound.Furthermore,for n?4,under the condition of the same curva-ture above,with the assumption of the first non-zero closed eigenvalue of Laplacian on the boundary,then the first non-zero Steklov eigenvalue comparison theorem is also true as same as the conclusion with n=2,3.These comparison theorems are the extensions of the Escobar's conclusions in[39].So we call the spectral comparison theorems of the first non-zero Steklov eigenvalues of the Laplace operator and the corresponding rigid-ity conclusions as the Escobar-type eigenvalue comparison theorems.In fact,the above Escobar-type eigenvalue comparison theorems are naturally important,which tell us that we can change the first non-zero steklov eigenvalue of the Laplace operator by changing the radial sectional curvature,and there is a rigidity characterization.Besides,we also give the lower bound estimates and the optimal upper bound esti-mates of the first non-zero Wentzell eigenvalues of weighted Laplace operator on smooth metric measure space.Specially,when the optimal upper bound is obtained,the domain is isometric to an Euclidean ball.2)Based on test function contructed in the proof of Escobar-type eigenvalue compar-ison theorems,and by the variational principle,we get a comparison theorem for the first nono-zero Wentzell eigenvalue of Laplacian on these manifolds with radial sectional cur-vature bounded above,and the rigidity conclusion can be characterized.Besides,we con-sider Reilly-type formula of weighted Laplacian,and use this formula to obtain a sharper lower bound for the Steklov eigenvalue problem of the weighted Laplaian on a compact smooth metric measure space with boundary and convex potention function,and this lower bound can be achieved only for the Euclidean ball of the prescribed radius,which gives a partial answer to the famous Escobar's conjecture proposed in[38].3)We obtain a Reilly-type integral inequality of the closed eigenvalue problem of weighted Laplacian on compact manifolds without boundary.Furthermore,when the outer space is sphere S^n+k,the rigidity conclusion can be achieved,which is a extension of the Du-Mao-Wang-Xia's conclusion([35,Theorem1]).