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Differential geometric researches on surfaces in a space of constant curvature and their singularities

Research Project

Project/Area Number 18540096
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo Denki University

Principal Investigator

KOKUBU Masatoshi  Tokyo Denki University, 工学部, 教授 (50287439)

Co-Investigator(Kenkyū-buntansha) IRIE Hiroshi  東京電機大学, 未来科学部, 講師 (30385489)
KOBAYASHI Shimpei  弘前大学, 大学院・理工学研究科, 助教 (40408654)
ROSSMAN Wayne  神戸大学, 大学院・理学研究科, 教授 (50284485)
Project Period (FY) 2006 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥3,700,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥600,000)
Fiscal Year 2009: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywords微分幾何 / 平均曲率 / ガウス曲率 / 特異点
Research Abstract

We studied surfaces in a three-dimensional manifold of constant negative curvature, called the hyperbolic space, requiring them to have good properties from the differential-geometric viewpoint. (Note that the hyperbolic space has interesting features beyond our common sense, e.g., a single hyperbolic line has infinitely many parallels.) We clarified the asymptotic behavior of ends of flat surfaces admitting singularities. Concerning linear Weingarten surfaces, we had a global representation formula, criterion for the shape of singularities, the orientability and co-orientability, and so on.

Report

(6 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report   Self-evaluation Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (28 results)

All 2009 2008 2007 Other

All Journal Article (13 results) (of which Peer Reviewed: 8 results) Presentation (15 results)

  • [Journal Article] Asymptotic behavior of flat surfaces in hyperbolic 3-space2009

    • Author(s)
      Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Journal of the Mathematical Society of Japan 61

      Pages: 799-852

    • NAID

      10026998469

    • Related Report
      2009 Annual Research Report 2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Surfaces and fronts with harmonic-mean curvature one in hyperbolic 3-space2009

    • Author(s)
      Masatoshi Kokubu
    • Journal Title

      Tokyo Journal of Mathematics 32

      Pages: 177-200

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] 3次元双曲型空間における平坦曲面のエンドの漸近挙動2009

    • Author(s)
      國分雅敏
    • Journal Title

      京都大学数理解析研究所講究録 1623

      Pages: 62-75

    • Related Report
      2009 Final Research Report
  • [Journal Article] Surfaces and fronts with harmonic-mean curvature one in hyperbolic three-space2009

    • Author(s)
      Masatoshi Kokubu
    • Journal Title

      Tokyo Journal of Mathematics 32

      Pages: 177-200

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 3次元双曲型空間における平坦曲面のエンドの漸近挙動2009

    • Author(s)
      國分 雅敏
    • Journal Title

      数理解析研究所講究録 1623

    • Related Report
      2008 Annual Research Report
  • [Journal Article] Asymptotics of ends of constant mean curvature surfaces with bubbletons2008

    • Author(s)
      Shimpei Kobayashi
    • Journal Title

      Proceedings of the American Mathematical Society 136

      Pages: 1433-1443

    • Related Report
      2009 Final Research Report 2008 Self-evaluation Report 2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 3次元双曲型空間の線形Weingarten曲面について2007

    • Author(s)
      國分雅敏
    • Journal Title

      京都大学数理解析研究所講究録 1527

      Pages: 49-66

    • Related Report
      2009 Final Research Report 2008 Self-evaluation Report
  • [Journal Article] On flat fronts in hyperbolic 3-space2007

    • Author(s)
      Masatoshi Kokubu
    • Journal Title

      Oberwolfach Report 24

      Pages: 1337-1340

    • Related Report
      2009 Final Research Report 2007 Annual Research Report
  • [Journal Article] Geodesics of Hofer's metric on the space of Lagrangian submanifolds2007

    • Author(s)
      Hiroshi Iriyeh
    • Journal Title

      Manuscripta mathematica 122

      Pages: 391-406

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms2007

    • Author(s)
      Shimpei Kobayashi
    • Journal Title

      The Journal of the London Mathematical Society 75

      Pages: 563-581

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 3次元双曲型空間の線形Weingarten曲面について2007

    • Author(s)
      國分 雅敏
    • Journal Title

      京都大学数理解析研究所講究録 No.1527

      Pages: 49-66

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Surfaces and fronts with harmonic-mean curvature one in hyperbolic three-space

    • Author(s)
      Masatoshi Kokubu
    • Journal Title

      Tokyo Journal of Mathematics (掲載決定)

    • Related Report
      2008 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of flat surfaces in hyperbolic 3-space

    • Author(s)
      Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Journal of the Mathematical Society of Japan (掲載決定)

    • NAID

      10026998469

    • Related Report
      2008 Self-evaluation Report
    • Peer Reviewed
  • [Presentation] 3次元双曲型空間のホロ球面的線型ワインガルテン曲面について2009

    • Author(s)
      國分雅敏
    • Organizer
      数理解析研究所研究集会「写像の特異点とそれに関連する幾何学」
    • Place of Presentation
      日本大学オーバルホール
    • Year and Date
      2009-12-09
    • Related Report
      2009 Final Research Report
  • [Presentation] 3次元双曲型空間のホロ球面的線型ワインガルテン曲面について2009

    • Author(s)
      國分雅敏
    • Organizer
      数理解析研究所 研究集会 '写像の特異点論とそれに関連する幾何学'
    • Place of Presentation
      日本大学オーバルホール
    • Year and Date
      2009-12-09
    • Related Report
      2009 Annual Research Report
  • [Presentation] 3次元双曲型空間の線形Weingarten曲面について2009

    • Author(s)
      國分雅敏
    • Organizer
      研究集会 '部分多様体幾何とリー群作用'
    • Place of Presentation
      東京理科大学森戸記念館
    • Year and Date
      2009-09-07
    • Related Report
      2009 Annual Research Report
  • [Presentation] Linear Weingarten surfaces in hyperbolic 3-space2009

    • Author(s)
      Masatoshi Kokubu
    • Organizer
      研究集会 'Differential Geometry with Mira'
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-04-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] 曲面の微分幾何---計算から眺めるいくつかの話題---2009

    • Author(s)
      國分 雅敏
    • Organizer
      研究集会'計算による数理科学の展開2009'
    • Place of Presentation
      神戸大学
    • Year and Date
      2009-01-09
    • Related Report
      2008 Annual Research Report
  • [Presentation] Asymptotic behavior of flat surfaces in H32008

    • Author(s)
      Masatoshi Kokubu
    • Organizer
      国際研究集会'Integrable systems, Geometry and Visualization
    • Place of Presentation
      九州大学箱崎地区・福岡大学
    • Year and Date
      2008-12-12
    • Related Report
      2009 Final Research Report
  • [Presentation] Asymptotic behavior of flat surfaces in H^32008

    • Author(s)
      Masatoshi Kokubu
    • Organizer
      国際研究集会'Integrable systems, Geometry and Visualization'
    • Place of Presentation
      九州大学
    • Year and Date
      2008-12-12
    • Related Report
      2008 Annual Research Report
  • [Presentation] On differential geometric aspect of flat fronts in hyperbolic 3-space2008

    • Author(s)
      Masatoshi Kokubu
    • Organizer
      9th International Workshop on Complex Structures, Integrability and Vector Fields
    • Place of Presentation
      Bulgarian Academy of Sciences, Sofia(ブルガリア)
    • Year and Date
      2008-08-26
    • Related Report
      2009 Final Research Report
  • [Presentation] On differential geometric aspect of flat fronts in hyperbolic 3-space2008

    • Author(s)
      Masatoshi Kokubu
    • Organizer
      9th International Workshop on Complex Structures, Integrability and Vector Fields
    • Place of Presentation
      Bulgarian Academy of Sciences, Sofia, Bulgaria
    • Year and Date
      2008-08-26
    • Related Report
      2008 Annual Research Report
  • [Presentation] 3次元双曲型空間における平坦曲面のエンドの漸近挙動2008

    • Author(s)
      國分雅敏
    • Organizer
      RIMS研究集会'部分多様体の微分幾何学およびその周辺領域の研究
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-06-24
    • Related Report
      2009 Final Research Report
  • [Presentation] 3次元双曲型空間における平坦曲面のエンドの漸近挙動2008

    • Author(s)
      國分 雅敏
    • Organizer
      RIMS研究集会'部分多様体の微分幾何学およびその周辺領域の研究'
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-06-24
    • Related Report
      2008 Annual Research Report
  • [Presentation] 曲面論におけるGauss写像の役割2008

    • Author(s)
      國分 雅敏
    • Organizer
      第3回札幌・福岡 幾何学セミナー
    • Place of Presentation
      北海道大学
    • Related Report
      2008 Annual Research Report
  • [Presentation] On flat fronts in hyperbolic 3-space2007

    • Author(s)
      Masatoshi Kokubu
    • Organizer
      Workshop 'Progress in Surface Theory
    • Place of Presentation
      Oberwolfach数学研究所(ドイツ)
    • Year and Date
      2007-05-02
    • Related Report
      2009 Final Research Report
  • [Presentation] On flat fronts in hyperbolic 3-space2007

    • Author(s)
      Masatoshi Kokubu
    • Organizer
      Workshop'Progress in Surface Theory'
    • Place of Presentation
      Oberwolfach数学研究所(ドイツ)
    • Year and Date
      2007-05-02
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] On flat fronts in hyperbolic 3-space2007

    • Author(s)
      Masatoshi Kokubu
    • Organizer
      Workshop 'Progress in Surface Theory'
    • Place of Presentation
      Oberwolfach数学研究所(ドイツ)
    • Year and Date
      2007-05-02
    • Related Report
      2007 Annual Research Report

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Published: 2006-04-01   Modified: 2016-04-21  

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