A Study On The Integral Geometry And Mean Curvature Flow Of Submanifolds

In this thesis, we mainly study the global geometry of submanifolds and the volume-preserving mean curvature flow of hypersurfaces in a space form. We obtain the rigidity theorem and topological sphere theorem of odd dimensional submanifolds in a space form, the rigidity theorem of submanifolds with parallel mean curvature in a unit sphere, the codimension reduction theorem of sub-manifolds in a unit sphere with parallel normalized mean curvature, the rigidity theorem of submanifolds in locally symmetric Riemannian manifolds, the con-vergence theorem of the volume-preserving mean curvature flow in a space form. This thesis consists of four parts.In chapter2, we study the rigidity theorem and topological sphere theorem of submanifolds in space forms under Ricci curvature pinching conditions. At the end of1960s, Simons、Lawson、Chern、do Carmo and Kobayashi proved the famous rigidity theorem of minimal submanifolds in the unit sphere. Later, Ejiri and Y.B.Shen obtained the rigidity theorem of minimal submanifolds in a unit sphere under Ricci curvature pinching conditions, Hai-Zhong Li improved the pinching constant for odd dimensional submanifolds. Recently, Xu and Gu generalized Ejiri’s rigidity theorem to submanifolds with parallel mean curvature in space forms. In this chapter, We prove:If M is an n(≥5)-dimensional compact oriented submanifold in Sn+P with parallel mean curvature, and if n is odd, RicM> C(n,p,H), then M is isometric to Sn(1/(?)), where C(n,p,H) is a positive constant depending on n,p and H. Our result improves Xu-Gu’s pinching constant for submanifolds of odd dimension. Using Lawson-Simons-Xin’s nonexistence theorem of stable currents, we prove:If M is an n(≥5)-dimensional compact submanifold in Fn+P(c) with c≥0, and if n is odd, RiCM>(n-2-∈n)(c+H2). then M is homeomorphic to a sphere, where∈n is a positive constant depending only on n. We also prove the topological sphere theorem for submanifolds in a hyperbolic space.In chapter3. we investigate the rigidity theorem of submanifolds with par- allel mean curvature in a unit sphere under scalar curvature pinching condi-tions. After Okumura, Yau and many other authors’ important works, H.W.Xu completely proved the generalized Simons-Lawson-Chern-do Carmo-Kobayashi theorem for submanifolds with parallel mean curvature in a sphere in1990. Using the DDVV inequality verified by Ge-Lu-Tang, we prove:If M is an n-dimensional compact oriented submanifold with parallel mean curvature in Sn+P. and if S+μ2≤α(n,H), where H(≠0) and S are the mean curva-ture and the squared norm of the second fundamental form of M respectively, then M is either the umbilical sphere he isoparametric hypersurface Sn-1(r1)×S1(r2) in a totally geodesic sphere Sn+1, the Clifford hypersurface S1(r3)×S1(r4) in the umbilical sphere S3(r), or the Veronese surface in the umbilical sphere S4(1/(?)). We also prove:If Mn is an n-dimensional compact oriented submanifold with parallel normalized mean curvature in Sn+P, and if H≠0, S+λ2<αα(n, H), then M lies in a totally geodesic sphere Sn+1.In chapter4, we investigate the rigidity of submanifolds with parallel mean curvature in a locally symmetric Riemannian manifold. In1995. H.W.Xu ini-tiated the study of minimal submanifolds in a general Riemannian manifold. Later, Shiohama and H.W.Xu proved the generalized Simons-Lawson-Chern-do Carmo-Kobayashi-Li-Li theorem for submanifolds with parallel mean curvature in a pinched Riemannian manifold. In this chapter, we prove:If Mn is an n-dimensional compact oriented submanifold with parallel mean curvature in a pinched locally symmetric Riemannian manifold, and if M’s Ricci curvature sat-isfies a given inequality, then M will be classified. Under sectional curvature conditions we obtain similar results for submanifolds in a locally symmetric Rie-mannian manifold.In chapter5, we investigate the convergence theorem of the volume-preserving mean curvature flow in a space form. In1987, Huisken studied the volume-preserving mean curvature flow of hypersurfaces in Euclidean space, and proved the convergence theorem for the volume-preserving mean curvature flow of uni-formly convex hypersurfaces in Euclidean space. At the same time, Gage in-vestigated the area-preserving mean curvature flow of curves in Euclidean plane. Later, Alikakos, Freire,Escher, Simonett, McCoy, Hao-Zhao Li and other authors investigated the volume-preserving mean curvature flow. Recently, Cabezas and Miquel proved the convergence theorem of hypersurfaces in a hyperbolic space Hn+P(c)(c<0) under piecewise curvature conditions. We prove the convergence theorem of the volume-preserving mean curvature flow of hypersurfaces in Sn+1or Hn+1under some curvature integral conditions.