Studies on Bernstein type theorems for minimal submanifolds
Project/Area Number 
11640068

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  IBARAKI UNIVERSITY (2000) University of Toyama (1999) 
Principal Investigator 
OKAYASU Takashi Ibaraki University, College of Education, Assistant Professor, 教育学部, 助教授 (00191958)

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: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)

Keywords  minimal submanifold / normal connection / equivariant differential geometry / hyperbolic space / 同変微分幾何 
Research Abstract 
Let G be a compact connected Lie group, Φ be an orthonormal representation of G on R^n with codimension 2 principal orbit type. Such (G, Φ, R^n) were classified completely by HsiangLawson (1971) and they are exactly those isotropy representations of symmetric spaces of rank 2. The orbit space R^n/G is a domain of R^2, it coinsides with the Weyl chamber R^2/W (W=Weyl group) of some symmetric space G/K.Hsiang (1982) reduced the problem of constructing hypersurfaces of constant mean curvature in R^n to solving the ordinaly differential equations of curves in R^2/W, and get many examples. In this research, using Hsiang's idea, we reduced the problem of constructing complete minimal submanifolds with flat normal connection in the Euclidean spaces, to solving some ordinaly differential equations of curves in R^2/W×R.The point is that submanifolds generated from rotating curves have always flat normal connection. Theorem 1 There are many codimension 2 irreducible complete minimal submanifolds with flat normal connection in the Euclidean spaces. Those examples are diffeomorphic to the following manifolds : S^P×S^q×R, SU (2)/T^2×R, G_2/T^2×R, F_4/Spin (8)×R,.... In 2000 using similar idea we get the following theorem. Theorem 2 There are many codimension 2 irreducible complete minimal submanifolds with flat normal connection in the hyperbolic spaces. Those examples are diffeomorphic to S^p×S^q×R.

Report
(3 results)
Research Products
(3 results)