Willmore Submanifold Geometry Rigid Eigenvalue Problem

In this thesis, we mainly study the geometric rigidity of Willmore submani-folds and extremal submanifolds in a sphere, the gap phenomenon for the eigen-values of Schrodinger operator on Willmore submanifolds and extremal subman-ifolds, and the spectrum of Laplace operator on extremal hypersurfaces.Willmore submanifolds and extremal submanifolds are two kinds of special submanifods in a sphere. They come from variation of two different functionals. Let M be an n-dimensional submanifold in the (n+p)-dimensional unit sphere Sn+p(1). Denote by and S the mean curvature and the squared length of the second fundamental form of M, respectively. We set andThe submanifold x:M→Sn+p is called a Willmore (or extremal) submanifold if it is an extremal submanifold to the functional W (or E, resp.). Li [Li1,Li3] and Guo [GL] investigate functionals W and E. They also obtain their Euler-Lagrange equations.In Chapter 2, we study the geometric rigidity of Willmore submanifolds. Li [Li1,Li2,Li3] proved a rigidity theorem under a pointwise pinching condition for Willmore submanifolds in a sphere. Using integral estimate method and Sobolev inequality, we obtain the global rigidity theorem for Willmore submanifolds in a sphere.Recently, Guo and Li [GL] obtained a rigidity theorem under a pointwise pinching condition for extremal submanifolds in a sphere. In Chapter 3, we prove a global rigidity theorem for extremal submanifolds, and a more beautiful pointwise one for extremal submanifolds with flat normal bundle. Using the idea in Yau's [Y1,Y2] for minimal submanifolds, we also get rigidity theorems of extremal submanifolds in a sphere under sectional and Ricci curvature pinching conditions.The eigenvalue of Schrodinger operator on Willmore submanifolds is our study object in Chapter 4. As early as in 1968, Simons investigated the first eigenvalue of one type Schrodinger operator on minimal submanifolds and ob-tained significant results. Later, Wu [Wu] obtained a more general result. We estimate the low bound of the first eigenvalue of Schrodinger operator on Will-more submanifolds and prove some rigidity theorems.We examine the relation between spectrum of Laplace operator on extremal hyper surf aces and the geometry in Chapter 5. Ding [D], Li and Wang [LW] studied cases for minimal hypersurfaces and Willmore hypersurfaces, respectively. We promote their results to extremal hypersurfaces.