Rigidity And Variational Problems In Geometry Of Submanifolds

Rigidity and variational problems are two kinds of important problems in geometry of submanifolds, which are widely studied by many geometers. Rigidity can be reflected by various pinching theorems. For a variational problem, we can study stability of the critical points and estimate the eigenvalues of the Jacobi operator. In this dissertation, we use a series of methods in geometry of submanifolds to study rigidity and stability of minimal submanifolds in Riemannian product manifolds, stability and eigenvalues of lin-ear Weingarten hypersurfaces in a sphere, and stable harmonic maps from a Riemannian product manifold to any Riemannian manifold. The main results are the following three parts:Firstly, we study the rigidity of compact minimal submanifolds in Sm(1)×R. We obtain a Simons’type equation, and then we prove some pinching theorems by the Ricci curvature and the sectional curvature pinching conditions, respectively. By our pinch-ing conditions, we conclude that, the minimal submanifolds lie in Sm(1)×{t0}=Sm(1). By the Ricci curvature pinching condition, we characterize the Clifford minimal hyper-surfaces. By the sectional curvature pinching condition, we characterize the Veronese submanifolds.Secondly, we show that linear Weingarten hypersurfaces in Sn+1(1) satisfying (n-1)H2+aH=b, where a and b are constants, can be characterized as critical points of the functional F=∫M (a+nH) dv for volume-preserving variations. We compute the first and second variational formulae of this functional, and prove that such a linear Weingarten hypersurface is stable if and only if it is totally umbilical and non-totally geodesic. This generalizes the stability results about hypersurfaces with constant mean curvature or with constant scalar curvature. We also obtain optimal upper bounds for the first and second eigenvalues of the Jacobi operator corresponding to the variational problem.Finally, we study stability of compact minimal submanifolds in Riemannian product manifolds. We prove a classification theorem for stable compact minimal submanifolds in M1×M2, where M1is an m1-dimensional (m1≥3) compact hypersurface in Euclidean space with the sectional curvature Km1satisfying1/(?)m1-1≤KM1≤1, and M2is any Rie-mannian manifold. This generalizes the result of Torralbo and Urbano for stable compact minimal submanifolds in Sm(r)×M. In particular, we prove that, when the ambient s- pace M is an m-dimensional (m≥3) compact hypersurface in Euclidean space with the sectional curvature satisfying1(?)m+1≤KM≤1, there exist no stable compact minimal submanifolds in M.