The First Stability Eigenvalues Of Hypersurfaces In Manifolds

As a generalization of Riemannian manifolds,weighted Riemannian manifolds are of great significance in the study of eigenvalue problems.In this paper,we study the first nonzero eigenvalue of the weighted stability operator on hypersurfaces immersed in weighted Reimannian manifolds,by applying the minimax principle and the divergence theorem,an upper bound of the first nonzero eigenvalue of the weighted stability operator is obtain.At the same time,the stability of the corresponding hypersurfaces is discuss,specifically includingFirstly,the weighted stability operator on hypersurfaces in weighted Reimanni-an manifolds is introduced and the corresponding first eigenvalue is define;Secondly,we study the closed hypersurfaces with constant weighted mean curvature in weight-ed Riemannian manifolds,and obtain the upper bound estimate of the first nonzero eigenvalue of the weighted stability operator under the pinching of the sectional curvature Sec?c,the upper bound is use to discuss the stability of hypersurfacesFinally,we study the closed hypersurfaces in weighted Reimannian manifolds,and obtain the upper bound estimates of the first nonzero eigenvalues of the weighted stability operator under the condition of Bakry-Emery-Ricci tensor Ricf pinching,some properties of this kind of hypersurfaces are discuss.