Submanifold Geometry, Analysis And Topology

In this thesis, we mainly study the geometry, analysis and topology of sub-manifolds. We obtain L^p-pinching theorems for locally conformally flat Riemannian manifolds, a geometric characterization of Clifford hypersurfaces, the classification of certain hypersurfaces in a 4-dimensional Hyperbolic space, topological and differentiable sphere theorems for Riemannian submanifolds, the convergence and extension over time for the mean curvature flow under integral curvature pinching conditions, etc.In chapter 2, we investigate the geometric rigidity for locally conformally flat Riemannian manifolds. M. Tani [T] showed that the universal cover of a compact oriented locally conformally flat manifold with positive Ricci curvature and constant scalar curvature is isometrically a sphere. Q. M. Cheng, S. Ishikawa and K. Shiohama [CIS] completely classified 3-dimensional complete and locally conformally flat Riemannian manifolds, whose scalar curvature and the norm of the Ricci curvature tensor are positive constants. Recently, S. Pigola, M. Rigoli and A. G. Setti [PRS] characterized a simply connected space form under a point-wise Ricci curvature pinching condition. They also proved a L₂ⁿ-pinching theorem for locally conformally flat Riemannian manifolds with zero scalar curvature. In Chapter 2, we prove some L^p-pinching theorems for locally conformally flat manifolds with constant scalar curvature, which improve and develop S. Pigola, M. Rigoli and A. G. Setti's L₂ⁿ-pinching theorem, and extend related results to the case where the scalar curvature is a nonzero constant.In Chapter 3, we study the characterization of Clifford hypersurfaces in the unit sphere. J. Simons[Si], H. Lawson[La], S. S. Chern, M. do Carmo and S. Kobayashi[CDK] proved a famous rigidity theorem for closed minimal submanifolds in the unit sphere. Basing on this result, Chern made a conjecture, known as Chern's conjecture, for minimal hypersurfaces with constant scalar curvature in a sphere. C. K. Peng, C. L. Terng[PT1], S. P. Chang[Cha1], H. C. Yang, Q. M. Cheng[YC1][YC2], Y. J. Suh and H. Y. Yang[SY], etc, have done some works on Chern's conjecture. Further, C. K. Peng and C. L. Terng[PT2] obtained the existence theorem of the second pinched interval for the scalar curvature of n-dimensional(n≤5) minimal hypersurfaces in a sphere. Recently, S. M. Wei and H. W. Xu[WX] extended C. K. Peng and C. L. Terng[PT2]'s result to the case where n = 6,7. In chapter 3, we investigate the existence of the second pinched interval for the scalar curvature of certain closed hypersurfaces with constant mean curvature in the unit sphere, which extends the existence theorem of the second pinched interval for the scalar curvature due to C. K. Peng, C. L. Terng, S. M. Wei and H. W. Xu, etc.In chapter 4, we investigate the geometric classification for hypersurfaces in a 4-dimensional Hyperbolic space. C. K. Peng and C. L. Terng[PT1] [PT2] studied the geometry of closed minimal hypersurfaces in a unit 4-sphere. S. Almeida, F. Brito, L. Sousa[AB1][ABS], J. Ramanathan[Ra], etc, studied the geometric classification for minimal hypersurfaces in a unit 4-sphere with constant Gauss-Kronecker curvature. T. Hasanis, A. Savas-Halilaj, T. Vlachos[HSV1][HSV2], Q. M. Cheng [Ch2], etc, investigated the classification of minimal hypersurfaces with constant Gauss-Kronecker curvature in 4 dimensional Euclidean space and Hyperbolic space. In this chapter, we obtain a classification of certain complete hypersurfaces with constant mean curvature and constant quasi-Gauss-Kronecker curvature in a 4-dimensional Hyperbolic space, which extends the results due to T. Hasanis, A. Savas-Halilaj, T. Vlachos and Q. M. Cheng, etc.In chapter 5, we prove topological and differentiable sphere theorems for Riemannian submanifolds. Using the nonexistence of stable currents on compact submanifolds in the sphere and the generalized Poincare conjecture, H. Lawson and J. Simons[LS] obtained the famous topological sphere theorem for submanifolds in a sphere. K. Shiohama and H. W. Xu[SX3]'s topological sphere theorem generalizes and improves H. Lawson and J. Simons' result. Recently, H. P. Fu and H. W. Xu[FX] extended the topological sphere theorem to the case where the ambient space is a Hyperbolic space. By studying the mean curvature flow, G. Huisken[Hu1][Hu2][Hu3] obtained some differentiable sphere theorems for certain hypersurfaces. In Chapter 5, we prove a topological sphere theorem for submanifolds in a general Riemannian manifold by investigating the isotropic curvature of submanifolds, which extends the topological sphere theorems for submanifolds in space forms to the case where the ambient space is a general Riemannian manifold. Using Brendle-Schoen's Ricci flow technique we initial the study of differentiable pinching problems for Riemannian submanifolds and obtain dif-ferentiable sphere theorems for submanifolds. Our pinching conditions in some sphere theorems are optimal.In chapter 6, we study the mean curvature flow on closed hypersurfaces in Euclidean space. G. Huisken [Hu1]proved that a convex hypersurface in the Euclidean space will shrink to a point in finite time along the mean curvature flow. He also obtained a sufficient condition to extend the mean curvature flow over time. Using a blowup argument, N. Sesum[Se] and B. Wang[W] recently proved the extension theorem for Ricci flow over time. In chapter 6, we prove several theorems of convergence and the extension over time for the mean curvature flow under integral curvature pinching conditions, which develop the works due to G. Huisken, N. Sesum, B. Wang, etc.