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Global analysis of curves and surfaces

Research Project

Project/Area Number 09304009
Research Category

Grant-in-Aid for Scientific Research (A)

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

Principal Investigator

KOISO Norihito  Grad. Sch. of Sci., Osaka University, Professor, 大学院・理学研究科, 教授 (70116028)

Co-Investigator(Kenkyū-buntansha) SAKANE Yusuke  Grad. Sch. of Sci., Osaka University, Professor, 大学院・理学研究科, 教授 (00089872)
NAMBA Makoto  Grad. Sch. of Sci., Osaka University, Professor, 大学院・理学研究科, 教授 (60004462)
USAI Sampei  Grad. Sch. of Sci., Osaka University, Professor, 大学院・理学研究科, 教授 (90117002)
YAMASAKI Yohei  Grad. Sch. of Sci., Osaka University, Associate Professor, 大学院・理学研究科, 教授 (00093477)
NISHITANI Tatsuo  Grad. Sch. of Sci., Osaka University, Professor, 大学院・理学研究科, 教授 (80127117)
梅原 雅顕  広島大学, 理学部, 教授 (90193945)
高橋 智  大阪大学, 大学院・理学研究科, 講師 (70226835)
竹腰 見昭  大阪大学, 大学院・理学研究科, 助教授 (20188171)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥27,400,000 (Direct Cost: ¥27,400,000)
Fiscal Year 1999: ¥8,000,000 (Direct Cost: ¥8,000,000)
Fiscal Year 1998: ¥7,500,000 (Direct Cost: ¥7,500,000)
Fiscal Year 1997: ¥11,900,000 (Direct Cost: ¥11,900,000)
Keywordsconstant mean curvature / stability / Einstein metric / 安定性 / 時空間図形
Research Abstract

We studied types of Jacobi fields of constant mean curvature surfaces. If the rotational surface is critical in the sense of stability, the Jacobi fields are usually rotationally symmetric. However, we found also rotationally non-symmetric one. The existence of non-symmetric Jacobi fields is not surprising, but the existence on the critical surface means that even symmetric stable solution of a variational problem may splits into non-symmetric solutions.
We proved that the hemi-sphere is the unique stable solutions of the free boundary problem of constant mean curvature surfaces with boundary in a plain. And, got a sufficient condition on the size of boundary and surface to be unique in the class of same boundary.
We classified homogeneous Einstein metrics on certain homogeneous spaces. In the classification, simplification of the algebraic equation and the efficiency of the algorism of solving algebraic equations are essential.
We prived that compact k-symmetric twister bundles over compact symmetric spaces are projected to compact symmetric spaces by primitive maps of finite type.
We got new estimates from below of the spectrum of general infinite graphs. The estimate is sharp on many cases.
We proved that global Sobolev-Bergman kernel can be analytically continued with respect to its Sobolev degree.
We generalized the theorem of Omori-Yau concerning the growth of the volume on complete riemannian manifold and the maximum principle.
We gave a sufficient condition to be strong solution of the weak solution of a boundary value problem with non constant rank of boundary matrix.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (18 results)

All Other

All Publications (18 results)

  • [Publications] H. Naiotoh: "Grassmann geometries on compact symmetrie spates of classical type"Japanese J. Math.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kato, K. and Usui, S.: "Logarithmic Htodge structures and their module (the fundamental diagram)"Proceedings of 「退化・被覆・特望点の代数幾何とトポロジー研究集会 1999」.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Itoh: "Weyl manztold and the Morse functional"Proceedings of the 4th Intern. Workship of Diff. Gear..

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Sasaki: "Projective differental geometry and linear homogeneous differential equations"Rokko Lectures In Math. 5. 115-115 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Shiohama & H. Xu: "Rigidity and sphere theorems for submanifolds II"Kyushu J. Math. 54. (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Koiso: "The uniqueness for stable surfaces of constant mean curvature with free boundary"Bulletin of Kyoto University of Education. 94. 1-7 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Naitoh: "Grassmann geometies on compact symmetric spaces of classical type"Japanese J. Math..

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kato, K. & Usui, S.: "Logarithmic Hodge structures and their moduli (The fundamental diagram)"Proceedings of 'Taika, Hifuku, Tokuiten no daisuukika to toporogii kenkyuushuukai 1999'.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Itoh: "Weyl Manifolds and the Morse Functional"Proceedings of the 4th Intern. Workshop of Diff. Geometry.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Sasaki: "Projective Differential Geometry and Linear Homogeneous Differential Equations"Rokko Lectures in Math.. vol. 5. 115-115 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Shiohama & H. Xu: "Rigidity and sphere theorems for submanifolds II"Kyushu J. Math.. vol. 54. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Koiso: "The uniqueness for stable surfaces of constant mean curvature with free boundary"Bulletin of Kyoto University of Education. vol. 94. 1-7 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Naitoh: "Grassmann geometies on compact symmetric spaces of classcal type"Japanese J. Math..

    • Related Report
      1999 Annual Research Report
  • [Publications] Kato, K. and Usui, S.: "Logarithmic Hodge structures and their moduli (The fundamental diagram)"Proceedings of 「退化・被覆・特異点の代数幾何とトポロジー研究集会 1999」.

    • Related Report
      1999 Annual Research Report
  • [Publications] M. Itoh: "Weyl Manifolds and the Morse Functional"Proceedings of the 4th Intern. Workshop of Diff. Geometry.

    • Related Report
      1999 Annual Research Report
  • [Publications] T. Sasaki: "Projective Differential Geometry and Linear Homogeneous Differential Equations"Rokko Lectures in Math.. 5. 115 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K. Shiohama and H. Xu: "Rigidity and sphere theorems for submanifolds II"Kyushu J. Math. 54. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] M. Koiso: "The uniqueness for stable surfaces of constant mean curvature with free boundary"Bulletin of Kyoto University of Education. B.94. 1-7 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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