Isometric immersions between spaces forms
Project/Area Number 
11640067

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Joetsu University of Education 
Principal Investigator 
MORI Hiroshi Joetsu Univ.of Edu., College edu., Prof., 学校教育学部, 教授 (00042185)

CoInvestigator(Kenkyūbuntansha) 
KUROKI Nobuaki Joetsu Univ.of Edu., College Edu., Prof., 学校教育学部, 教授 (70059731)
TANAKA H. Joetsu Univ.of Edu., College Edu., Prof., 学校教育学部, 教授 (10033846)
MATSUMOTO Kengo Joetsu Univ.of Edu., College Edu., Asso.Prof. (40241864)

Project Period (FY) 
1999 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2000: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1999: ¥1,400,000 (Direct Cost: ¥1,400,000)

Keywords  rotation hypersurfaces / mean curvature / hyperbolic space / stable hypersurfaces / notation hypersurface / staffe hypersurface / C^*algebra / subshift / dimesion group 
Research Abstract 
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 oneparameter 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.

Report
(3 results)
Research Products
(4 results)