Some Rigidity Results Of Submanifolds With Constant Mean Curvature

In this thesis, we consider the Bochner type formula on the submanifold in Euclidean space, then get the Bochner type inequality on the hypersurface with constant mean curvature in Euclidean spaceand the Bochner type inequality on the higher codimension minimal submanifold in Euclidean spaceAs we know there is a support function w on constant mean curvature hypersur-face(in fact, for submanifold with parallel mean curvature in Euclidean space) satisfying a famous equationAccording to this support function and the above equation, we obtain a subhar-monic function on the hypersurface. Then the interior gradient estimate leads to following result定理0.3:[L-M] Let M be a complete constant mean curvature hypersurface of dimension n≤3 with positive support function w in Rⁿ⁺¹,and V =(?).Ifwhere r is the Euclidean distance from a fixed point in M. Then M has to be an affine linear subspace. Secondly, as for minimal submanifold, we use the Bochner type inequality, combining the L² - Sobolev inequality引理0.3:Let Mⁿ be a minimal submanifold in R^n+p,then there exists S(n) =(?) such thatfor any compact supported smooth functionφon M.Then we can get the following result定理0.4: [L] Let Mⁿ(n≥3) be a complete minimal immersed submanifold in R^n+p. If there exists a constant C₁(n)=(?) > 0 such thatthen M must be totally geodesic. Here B is the second fundamental form of Mⁿ and S(n) is the constant in the L² -Sobolev inequality.At last, we generalize a result in [S-Z] to minimal submanifold, by some modification,we find their method can also be used. Then we get the following result命题0.4: [L] Let M be a proper minimal submanifold in R^n+p with boundaryδM (may be empty). Letα₁,α₂(0<α₁<α₂) be two real number such thatδM∩BT(0,α₁,α₂) =φ,then there exists a uniform constant ##₀ > 0 depending only on n such that ifthenwhere, a_m=(?),|B| denotes the norm of the second fundamental form on M.