Geometry of Total Curvature on Negatively Curved Manifolds
Project/Area Number  09640099 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  Toyama University 
Principal Investigator 
OKAYASU Takashi Toyama University, Faculty of Education, Assistant Professor, 教育学部, 助教授 (00191958)

CoInvestigator(Kenkyūbuntansha) 
笹尾 靖也 富山大学, 教育学部, 教授 (10016024)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥2,400,000 (Direct Cost : ¥2,400,000)

Keywords  Bernstein theorem / higher codimensional graph / higher order mean curvature / halfspace theorem / normal connection / 余次元の高いグラフ / 高次平均曲率 / half space theorem / 最大値原理 / Bennstein Problem / 極小部分多様体 
Research Abstract 
In 1997 we studied how the total curvature changes through the mean curvature flow by using the method of Hamilton and Huisken. As a byproduct, we got a Bernstein type thorem for minimal submanifolds in the Euclidean space. Theorem 1 Suppose that u=(u^1, ..., u^p) : R^n*R^p satisfies the system of minimal surface equation and its graph graph(u) has flat normal connection. If <<numerical formula>> then all u^i are linear functions. In 1998, we extended the halfspace theorem for minimal hypersurfaces by Hoffman and Meeks (1990) to hypersurfaces with 0 higher order mean curvature. Let M^n * R^<n+1> be a hypersurface. We define kth mean curvatue H_k by H_k= SIGMA__<i_1<...<i_k> lambda_1 ... lambda_<ik>, where lambda_1, ..., lambda_n are principal curvatures of M.Suppose k is odd. We call M elliptic type if the following condition holds everywhere : */(mbda) H_k > 0 for *i. Note that this condition does not depend on the choice of the unit normal vector since k is odd. Theorem 2 Let k be an odd integer, n an ineger satisfying 1 <less than or equal> k < n <less than or equal> 2k. If M^n * R^<n+1> is a properly immersed elliptic type complete hypersurf ace with H_k = 0, then M cannot be contained in any Euclidean halfspace.

Report
(4results)
Research Output
(6results)