Studies On The Stability,Rigidity And Eigenvalue Estimate Of Submanifolds In Riemannian Space

In this paper,we mainly study the stability and rigidity of the linear Weingarten hypersurface in Riemannian space form and the application of stable hypersurface in general Riemannian space.We also give the estimate of the first eigenvalue of the weighted p-Laplacian in hyperbolic space.The paper consists of three parts(Chapter 3 to Chapter 5).In Chapter 3,we show some results about hypersurfaces in space forms.Firstly,we prove some stability results about linear Weingarten hypersurfaces in space forms,which generalize the stability results about the hypersurfaces with zero mean curvature or with zero scalar curvature in[4].Secondly,we give some stability results about hypersurfaces in S4 and H4,which generalize the stability results in[58].Finally,we get a vanishing theorem for harmonic forms on complete Riemannian manifolds.In Chapter 4,we deal with linear Weingarten submanifolds in a Riemannian space form with parallel normalized mean curvature vector.With some limitations,we obtain a rigidity theorem by modifying Cheng-Yau's technique,which is a generalization of the result in[2].In Chapter 5,a lower bound estimate for the first eigenvalue of weighted p-Laplace operator on submanifolds in hyperbolic space is given.