Variational Problems And Rigidity Theorems Of Submanifolds In Riemannian Manifold

Variational theory of submanifolds is an important branch of Rimannian geometry.In this thesis,we study the variational problems of submanifolds theory.The main results are the following four parts:Firstly,let x:M→Sn+p(1)be an n-dimensional submanifold immersed in an(n+p)-dimensional unit sphere Sn+p(1).We study n-dimensional submanifolds immersed in Sn+p(1)which are critical points of the functional S(x)=∫MSn/2 dv,where S is the squared length of the second fundamental form of the immersion x.When x:M→ S2+p(1)is a surface in S2+p(1),the functional S(x)=∫MSn/2 dv represents double volume of image of Gaussian map.The critical surface of S(x)are S-surfaces,we proved an integral inequality of Simons’ type for compact Ssurface in S2+p(1).Furthermore,we establish a rigidity theorem for the S-surface in the unit sphere S2+P(1).Secondly,let H be the mean curvature vector of an n-dimensional submanifold in(n+p)-dimensional unit sphere Sn+p(1).The functional H(x)=∫M‖H‖ndv is called the total mean curvature functional.We present the first variational formula of H(x)and then,for a critical surface of H(x)in the(2+p)dimensional unit sphere S2+p(1),we establish the relationship between an extrinsic quantity of the surfaces and its Euler characteristic.Next,we study the M?bius minimal and M?bius isotropic hypersurfaces in the unit sphere S5(1).The M?bius minimal hypersurfaces are the critical points of M?bius volume functional which corresponds to Willmore hypersurfaces in the sphere.We prove that a M?bius minimal and M?bius isotropic hypersurfaces of the unit sphere S5(1)is M?bius equivalent to either the Clifford torus or the Cartan minimal hypersurface with four distinct principal curvatures.At last,we study the locally conformally flat real hypersurfaces in both the n-dimensional complex quadric Qn and the complex hyperbolic quadric Qn*for n≥3,we prove the non-existence of locally conformally flat real hypersurfaces in both the n-dimensional complex quadric Qn and the complex hyperbolic quadric Qn*for n≥ 3 by applying for a new approach of the so-called Tsinghua principle.