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关于常曲率空间形式中子流形的一些结果
Results of the Submanifolds in the Space Form with Constant Curvature
【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。
【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.
【Key words】 space with constant curvature; the second fundamental form; Simons-type inequality; stable currents; principal curvature;
- 【网络出版投稿人】 西北师范大学 【网络出版年期】2008年 07期
- 【分类号】O186.1
- 【下载频次】47