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Studies on Bernstein type theorems for minimal submanifolds

Research Project

Project/Area Number 11640068
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionIBARAKI 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)
Keywordsminimal 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 Hsiang-Lawson (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)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (3 results)

All Other

All Publications (3 results)

  • [Publications] 岡安隆: "A Remark on Stable Complete Minimal Hypersurfaces in Enclidean Space"Mathematical Journal of Toyama University. Vol.23. 77-78 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takashi Okayasu: "A remark on stable complete minimal hypersurfaces in Euclidean Space"Mathematical Journal of Toyama University. vol. 23. 77-78 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 岡安隆: "A Remark on Stable Complete Minimal. Hypersurfaces in Euclidean Space"Mathematical Journal of Toyama university. Vol.23. 77-78 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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