Relationship Between Eigenvalue Of Jacobian Operator And Laplace Operator For Hypersurfaces In Riemannian Space

In this paper,we mainly study the relationship between the eigenvalue of Jacobian operator and Laplace operator of constant mean curvatures with free boundary hypersur-faces in sphercial and Euclidean space.We also give the proof of the relationship between the eigenvalue of weighted f-Jacobian operator and f-Laplace operator of constant f-mean curvatures hypersurfaces.The paper consists of two parts(Chapter 3 and Chapter 4).In chapter 3,inspired by[41],we study the relationship between the eigenvalue of Jacobian operator and Laplace operator of constant mean curvatures hypersurfaces in sphercial and Euclidean space.We generalize the related results of minimal hypersurfaces in sphercial spaces to constant mean curvature hypersurfaces.In chapter 4,we estimate the eigenvalue of weighted f-Jacobian operator and f-Laplace operator of constant f-mean curvatures hypersurface with free boundary in weighted sphercial spaces and weighted Euclidean spaces.We generalize[24]the results of f-minimal hypersurfaces in weighted Euclidean spaces to f-mean curvatures hypersur-faces.