Rigidity Of Submanifolds And Problem Of Eigenvalue

In this thesis, we study the rigidity of submanifolds and problem of eigenvalue. We divide the study into four parts.First, we study the rigidity of compact minimal submanifolds in locally symmetric spaces. Yau.S.T studied the compact minimal submanifolds in constant curvature spaces and obtained results similar with simons'inequality[1]. We consider the minimal submanifolds in the locally symmetric Riemannian manifold, taking the values of any real number a, we obtain some of the rigidity theorem, have been the inequalities of the square of the length and cross-sectional curvature, extend Yau's results to the locally symmetric space, proof of the followingTheorem 2.1 Let Nn+p be a (n+p)-dimensional locally symmetric Riemannian manifold with sectional curvature KN satisfying 0<δ≤KN≤1, Mn be a compact minimal submanifolds in N"+p, and s be a square of the length of the second fundamental form in Mn. Let k is a function of M", For any point x∈M",k(x) is the infimum of the arbitrary two-dimensional sectional curvature. If for any real number have Then the second fundamental form of Mn is covariant constant, and Mn is a totally geodesic submanifold and locally symmetric, otherwise Particularly, we give a= 1, obtain the followingCorollary Under the conditions in Theorem 2.1, if s satisfies Then the second fundamental form of Mn is covariant constant, and Mn is a totally geodesic submanifold and locally symmetric, otherwise Note whenδ= 1, N"+p be a constant curvature space with KN= 1, at this point was exactly the case of c= 1 at [1].Theorem 2.2 Under the conditions in Theorem 2.1, let a=(p-1)/p, if the sectional curvature RM of Mn satisfies (i) if s= 0, Mn is a totally geodesic submanifold and locally symmetric, otherwise sectional curvature RM.Second, we study the classification of the semi-parallel hypersurfaces in a anti-de Sitter space H1 n+1 (-1) and the equivalence between the hypersurfaces with parallel higher order and the parallel hypersurfces in H1 n+1 (-1). F.Dillen gave the semi-parallel hypersurfaces classification in Euclidean space[6],▽kh= 0 and Vh= 0 are non-equivalent in Euclidean space, but for the constant curvature space Nn+1(c), zhang ting-fang proved the equivalent of▽kh= 0 and▽h= 0, and the classification of the semi-parallel hypersurfaces in the constant curvature space[23]. We extend this conclusion to anti-de Sitter space H1 n+1 (-1) as followingTheorem 3.1 Let M" be a space-like hypersurface in a anti-de Sitter space H1 n+1(-1), if n> 2, and Mn is semi-parallel, then Mn is the open part of the hypersurface of one of the following(i) totally umbilical hypersurface in H1 n+1 (-1);(ii) two constant curvature submanifolds of the product in H1 n+1(-1);(iii) (n-1)-dimensional totally umbilical submanifold Mn-1(υ) along the trajectory path with orthogonal movement in H1 n+1 (-1),υbe the arc-length parameter of orthogonal trajectory path, and n-1-remain curvatureλalong Mn-1(υ) is a constant, select the appropriate initial parameters, principal curvatureλhas using the above classification theorem, we prove the equivalence between the parallel hypersurfaces with higher order and the parallel hypersurfces in H1 n+1(-1).Theorem 3.2 Let Mn be a space-like hypersurface in a anti-de Sitter space H1 n+1 (-1), then▽kh=0 and▽h= 0 are equivalent.Third, we study the eigenvalues of Schrodinger operator and Laplace operator for the Dirichlet problem. We establish the following inequalities to estimate the (k+1)-th eigenvalueλk+1 in terms of the first k eigenvalues in these two cases.Theorem 4.1 LetΩbe a connected bounded domain in sphere Sn(1), n is the unit outward normal to (?)Ω. Assume that are eigenvalues of Schrodinger operator for the Dirichlet problem in which W is a continuous function of Sn(1),thenTheorem 4.2 LetΩbe a connected bounded domain in an n-dimensional Euclidean space Rn, n is the unit outward normal to (?)Ω. Assume that are eigenvalues of Laplacian operator with any order l for the Dirichlet problem then When l= 1, there This is the same result reported by [12], our results are its high-order extension.Forth, we study the rigidity of non-negative Ricci curvature compact Riemannian manifold and obtain the estimate of the first non-empty boundary eigenvalue. In this paper, we improve the results using Reilly formula to establish the following theoremTheorem 5.1 LetΩbe a n+1-dimensional compact Riemannian manifold with boundary of the Ricci Curvature lower bound for k≥0, at the non-empty boundary M= (?)Ω, we have induced metric ofΩ, then When k= 0,λ1(M)≥nc2 is the same result reported by [10].