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Isometric immersions between spaces forms

空间形式之间的等距沉浸

基本信息

批准号：
11640067
负责人：
MORI Hiroshi
金额：
$ 2.18万
依托单位：
Joetsu University of Education
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

Hypersurfaces M^n with constant mean curvature in a Riemannian manifold M^^〜^<n+1> are solutions to the variational problem of minimizing the area function for certain variations ; the admissible variations are only those that leave a certain volume function fixed. This isoperimetric character of the variational problem associated to hypersurfaces with constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces.There are many complete hypersurfaces with constant mean curvature in Euclidean (n+1)-space R^<n+1> and Euclidean (n+1)-sphere S^<n+1>, but in the hyperbolic (n+1)-space H^<n+1> there have been few results on such hypersurfaces except umbilical ones. First main purpose of this paper is to construct one-parameter families of three distinct type, rotation hypersurfaces with constant mean curvature in H^<n+1>, explicitly.Barbosa, do Carmo and Eschenburg have defined the notion of stability for hypersurfaces M^n with constant mean curvature in a Riemannian manifold M^^〜^<n+1>. The case where M^2 is complete and noncompact is treated by da Silveira. The case where M^n, is compact is treated by Barbosa, do Carmo and Eschenburg. Luo has discussed the stability of complete noncompact hypersurfaces with constant mean curvature in R^<n+1>.Except for the case where H=0 very little is known about stability of complete and noncompact Riemannian hypersurfaces of H^<n+1> with constant mean curvature H, when 3【less than or equal】n. Second main purpose of this paper is to discuss the stability of the hypersurfaces in H^<n+1> with constant mean curvature H.

黎曼流形M^^ ~ ^<n+1>中具有常平均曲率的超曲面M^n是对面积函数在一定变化下最小化的变分问题的解；允许的变化只是那些使某个体积函数保持固定的变化。与具有常平均曲率的超曲面相关的变分问题的这种等周特性在处理此类超曲面的稳定性时引入了额外的复杂性。在欧氏(n+1)-空间R^<n+1>和欧氏(n+1)-球面S^<n+1>中有许多具有常平均曲率的完全超曲面，但在双曲(n+1)-空间H^<n+1>中，除了脐带型超曲面外，几乎没有关于这类超曲面的研究结果。本文的第一个主要目的是明确地构造三种不同类型的单参数族，H^<n+1>中具有恒定平均曲率的旋转超曲面。Barbosa， do Carmo和Eschenburg定义了黎曼流形M^^ ~ ^<n+1>中具有常平均曲率的超曲面M^n的稳定性概念。M^2是完全非紧的情况由da Silveira处理。M^n是紧致的情况由Barbosa， do Carmo和Eschenburg处理。Luo讨论了R^<n+1>范围内具有常平均曲率的完全非紧超曲面的稳定性。除了H=0的情况外，对于H^<n+1>且平均曲率H为常数的完备非紧黎曼超曲面，当3 <或等于n时，其稳定性知之甚少。本文的第二个主要目的是讨论H^<n+1>中具有恒定平均曲率H的超曲面的稳定性。