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Studies on minimal surfaces and Simon conjecture

Research Project

Project/Area Number 10640063
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionSaitama University

Principal Investigator

SAKAMOTO Kunio  Saitama University, Faculty of Science Professor, 理学部, 教授 (70089829)

Co-Investigator(Kenkyū-buntansha) NAGASE Masayosi  Saitama University, Faculty of Science Professor, 理学部, 教授 (30175509)
OKUMURA Masafumi  Saitama University, Faculty of Science Professor, 理学部, 教授 (60016053)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2000: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1999: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1998: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsMinimal surface / Simon conjecture / Willmore surface / normal curvature tensor / Variational problem / Concircular scalar field / 楕円関数 / 曲率 / ガウス曲率
Research Abstract

U.Simon conjectured in 1980 that, for a closed connected minimal surface immersed in an n-dimensional unit sphere, if the Gauss curvature is greater than or equal 2/n(n+1) and less than or equal 2/n(n-1), then it is a surface of constant curvature. We partially solved this conjecture and published the article from Math. Zeit.. In this paper, we proved that if the Laplacian of the Gauss curvature is pinched by certain quadratic polynomials, then the conjecture is true. Moreover, we studied the degeneracy of the higher order normal space in the case that the Gauss curvature is greater than or equal 1/8 and less than or equal 1/6. Also we obtaied a result that if the ratio of the metric induced by the directrix curve and the induced one on the surface is less than or equal to three times the Gauss curvature, then the surface is a standard constant curvature sphere and an inequality which shows that the greater the degree of the degeneracy becomes the ratio becomes greater. Since 1999, we … More studied conformal invariants concerning the length of the normal curvature tensor for submanifolds immersed in a space of constant curvature. We obtained the first variation formula for some variational problem and studied the properties of the critical surfaces. In particular, the result that if the normal connection of a 4-dimensional compact submanifold is self-dual or anti-self-dual, then it is critical was shown. We also considered 2-dimensionl cases. Under the condition that the length of the normal curvature tensor is a nonzero constant and the curvature ellipse is a circle, critical surfaces are Willmore one and vica versa. Concerning this result, we had a logarithmic residue formula about the S-Willmore points of a Willmore surface, especially, represented the Willmore integral of a Willmore sphere immerced in a 6-dimensional sphere by the logarithmic residue and the Euler number. Moreover, we proved that if a compact critical surface satisfies the condition that the mean curvature normal is parallel and curvature ellipse is a circle, then it is of constant curvature. In the proof of this result, making use of elliptic functions, we classified surfaces admitting a concircular scalar field with characteristic function of degree 2 or 3 and applied this to the proof. The article is submitted to a journal. Less

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (15 results)

All Other

All Publications (15 results)

  • [Publications] K.Sakamoto: "On the curvature of minimal 2-spheres in spheres"Math.Zeit.. 228. 605-627 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Okumura: "CR submanifolds of maximal CR dimension of complex projective space"Bull.Greek Math.Soc.. 44. 31-39 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Djorc and M.Okumura: "On contact submanifolds in complex projective space"Math.Nachr.. 202. 17-28 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Djorc and M.Okumura: "CR submanifolds of maximal CR dimension of complex projective space"Arch.Math.. 71. 148-158 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Nagase: "Twistor space and Seiberg-Witten equation"Saitama Math.J.. 18. 39-60 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Sakamoto: "On the curvature of minimal 2-spheres in spheres"Math.Zeit.. 228. 605-627 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Okumura: "CR submanifolds of maximal CR dimension of complex projective space"Bull.Greek Math.Soc.. 44. 31-39 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Djorc and M.Okumura: "On contact submanifolds in complex projective space"Math.Nachr.. 202. 17-28 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Djorc and M.Okumura: "CR submanifolds of maximal dimension of complex projective space"Arch.Math.. 71. 148-158 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Nagase: "Twistor space and Seiberg-Witten equation"Saitama Math.J.. 18. 39-60 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kunio Sakamoto: "On the curvature of minimal 2-spheres in spheres"Math.Zeit.. 228. 605-627 (1998)

    • Related Report
      2000 Annual Research Report
  • [Publications] Masayoshi Nagase: "Twister space and Seiberg-Witten equation"Saitama Math.J.. 18. 39-60 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kunio Sakamoto: "On the curvature of minimal 2-spheres in spheres"Math. Z.. 228. 605-627 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] 阪本 邦夫: "On the curvature of minimal 2-spheres in spheres" Math.Z.228. 605-627 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 奥村 正文: "CR submanifold of maximal CR dimension of complex projective space" Arch.Math.71. 148-158 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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