【作者】 张秋燕；

【导师】 刘建成；

【作者基本信息】 西北师范大学 ， 基础数学， 2007， 硕士

【摘要】 本文主要讨论常曲率空间形式中的子流形,得到以下结果:1.对于单位球面中的紧致子流形,得到了第二基本形式模长平方S关于其上Laplacian算子第一非零特征值的SimonS型不等式;进而,在S为常数的假设下,得到S的下界估计式。2.对于欧氏空间中的紧致子流形,得到了某些类子流形稳定流的非存在性结果。关于超曲面,我们在假设这些类子流形的主曲率,截面曲率或者第二基本形式模长平方分别满足某种条件下,证明了相应的非存在性定理。关于余维数大于1的情形,我们也证明了,如果子流形的第二基本形式模长平方满足某一拼挤条件,那么该子流形中不存在稳定流,而且这类子流形与欧氏球同胚。3.对于双曲空间中的子流形,讨论了双曲空间中具有非正Ricci曲率超曲面的性质,得到了超曲面第二基本形式模长平方的一个下界。进而,得到了超曲面主曲率乘积的一个上界。更多还原

【Abstract】 The aim of this paper is to deal with the submanifolds immersed into the space form with constant curvature and obtain the following results:1. For the compact submanifolds of the standard Euclidean sphere, we shall give a Simons-type inequality involving the squared norm S of the second fundamental form h of Mⁿ in terms of the first nonzero eigenvalueλ₁ of the Laplacian of Mⁿ. Furthermore, if, in addition, S is a constant, we give the lower bound for S.2. For the compact submanifolds of the Euclidean space, we shall prove the non-existence of stable currents for certain classes of them. For hypersurfaces, we prove the non-existence theorems under the assumptions about the principal curvature, sectional curvature or the square length of the second fundamental form respectively. For high-codimension, we also prove that there are no stable currents in submanifolds of the Euclidean space when the square length of the second fundamental form satisfies a pinching condition. As a result, such submanifolds are homeomorphic to the Euclidean sphere.3. For the submanifolds of the hyperbolic space, we consider the hypersurfaces with non-positive Ricci curvature and give a lower bound for the square length of the second fundamental form of the hypersurfaces. Further, we obtain an upper bound for the product of the principal curvatures of the hypersurfaces.更多还原