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Energy of knots and conformal geometry

Research Project

Project/Area Number 15540088
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

OHNITA Yoshihiro (2004)  Tokyo Metropolitan University, Dept. of Math., Professor, 理学研究科, 教授 (90183764)

今井 淳 (2003)  東京都立大学, 理学研究科, 助教授 (70221132)

Co-Investigator(Kenkyū-buntansha) IMAI Jun  Tokyo Metropolitan University, Dept. of Math., Associate professor, 理学研究科, 教授 (70221132)
OKA Mutsuo  Tokyo Metropolitan University, Dept. of Math., Professor, 理学研究科, 教授 (40011697)
横田 佳之  東京都立大学, 理学研究科, 助教授 (40240197)
KOJIMA Sadayoshi  Tokyo Institute of Technology, Professor, 情報理工学研究科, 教授 (90117705)
AHARA Kazushi  Meiji University, Lecturer, 理工学部, 講師 (80247147)
GUEST Martin  Tokyo Metropolitan University, Dept. of Math., Professor (10295470)
神島 芳宣  東京都立大学, 理学研究科, 教授 (10125304)
大仁田 義裕  東京都立大学, 理学研究科, 教授 (90183764)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
Keywordsknot / energy functional / conformal geometry / エネルギー / 共形幾何 / メビウス変換 / 非調和比
Research Abstract

Let Y and Y' be a pair of curve segments in an n dimensional sphere, and let x, x+dx be points on Y and y, y+dy be points on Y'. We allow the case when Y and Y' coincide, when we always assume that x and y are distinct. By identifying a 2 dimensional sphere through the four points x, x+dx, y, and y+dy with the Riemann sphere through a stereographic projection, we obtain the cross ratio of these four points. Then it can be considered as a complex valued 2-form on YxY'. We call it the infinitesimal cross ratio of Y and Y'. It is, by definition, invariant under Moebius transformations.
We obtained new interpretations of the real and the imaginary parts of the infinitesimal cross ratio.
An n dimensional sphere can be realized as the set of points a infinity of the light cone in (n+2) dimensional Minkowski space. Let S(n, p) denote the set of p dimensional sphere in the n dimensional sphere. Then S(n, p), which can be expressed in terms of Pluecker coordinates, is a space with an indefinite m … More etric. A pair of curve segments Y and Y' in the n dimensional sphere can also be considered as a surface in S(n, O). Now the real part of the infinitesimal cross ratio is equal to the absolute value of the area element of this surface.
On the other hand, the interpretation of the imaginary part can be given as follows. An n dimensional sphere can be considered as the boundary of the (n+1) dimensional hyperbolic space. Let L denote a geodesic in the hyperbolic space joining points x on Y and y on Y' in the boundary sphere, and let P be an orthogonal hyperplane to L. Let x' and y' be points in neighborhoods of x and y respectively. The intersection of P and the geodesic joining x' and y' gives a surface in P. Then the imaginary part of the infinitesimal cross ratio at (x, y) is equal to the area element of this surface.
We also defined functionals on the space of curves and surfaces using a conformally invariant measure on the space S(n, p).
(That is the summary of the joint work of Jun Imai, who was the head investigator in 2003, and Remi Langevin, who is an investigator abroad, during Imai's 7 months stay in France in 2004. Part of the grant was used to invite Lengevin to Japan in 2003.) Less

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (22 results)

All 2005 2004 2003 Other

All Journal Article (18 results) Book (2 results) Publications (2 results)

  • [Journal Article] Conformally invariant energies of knots2005

    • Author(s)
      R.Langevin, J.O'Hara
    • Journal Title

      J.Institut Math.Jussieu 4

      Pages: 219-280

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Quantum cohomology via D-modules2005

    • Author(s)
      M.Guest
    • Journal Title

      Topology 44

      Pages: 263-281

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Conformally invariant energies of knots2005

    • Author(s)
      Langevin, R., O'Hara, J.
    • Journal Title

      J. Institut Math. Jussieu 4

      Pages: 219-280

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Quantum cohomology via D-modules2005

    • Author(s)
      Guest, M.
    • Journal Title

      Topology 44

      Pages: 263-281

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Conformally invariant energies of knots2005

    • Author(s)
      J.O'Hara
    • Journal Title

      J.of the Inst.of Math.Jussieu 4(2)

      Pages: 219-280

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On topological types of reduced sextics2004

    • Author(s)
      M.Oka
    • Journal Title

      Kodai J.Math. 27

      Pages: 237-260

    • NAID

      130003574463

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A formula for the A-polynomial of (-2,3,1-2n)-pretzel knots2004

    • Author(s)
      Y.Yokota
    • Journal Title

      Tokyo J.Math. 27

      Pages: 263-273

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] The Dehn filling space of a certain hyperbolic 3-orbifold2004

    • Author(s)
      S.Kojima, S.Mizushima
    • Journal Title

      Contemporary Math. 347

      Pages: 131-140

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On topological types of reduced sextics2004

    • Author(s)
      Oka, M.
    • Journal Title

      Kodai J. Math. 27

      Pages: 237-260

    • NAID

      130003574463

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A formura for the A-polynomial of (-2,3,1-2n)-pretzel knots2004

    • Author(s)
      Yokota, Y.
    • Journal Title

      Tokyo J. Math. 27

      Pages: 263-273

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The Dehn filling space of a certain hyperbolic 3-orbifold2004

    • Author(s)
      Kojima, S., Mizushima, S.
    • Journal Title

      Contemporary Math. 347

      Pages: 131-140

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On topological types of reduced sextics2004

    • Author(s)
      Y.Oka
    • Journal Title

      Kodai J.Math 27

      Pages: 237-260

    • NAID

      130003574463

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Dehn filling space of a certain hyperbolic 3-orbifold2004

    • Author(s)
      S.Kojima
    • Journal Title

      Contemporary Math. 347

      Pages: 131-140

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Energy of knots and conformal geometry2003

    • Author(s)
      O'Hara, J.
    • Journal Title

      Series on Knots and Everything, World Scientific, Singapore Vol.33

      Pages: 304-304

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Sphairahedral Approach to Parameterize Visible2003

    • Author(s)
      Kojima, S., Mizushima, S., Ahara, K., Araki, Y.
    • Journal Title

      Three Dimensional Quasi-Fuchsian Fractals, CG12003

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Gauge-theoretic approach to harmonic maps and subspaces in muduli spaces

    • Author(s)
      Y.Ohnita
    • Journal Title

      "Integrable Systems, Geometry and Topology" NCTS volume, International Press 発表予定

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Gauge-theoretic approach to harmonic maps and subspaces in muduli spaces

    • Author(s)
      Ohnita, Y.
    • Journal Title

      Integrable Systems, Geometry and Topology NCTS volume, International Press (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Gauge-theoretic approach to harmonic maps and subspaces in moduli spaces

    • Author(s)
      Y.Ohnita
    • Journal Title

      Integrable Systems, Geometry and Topology (NCTS volume, International Press) (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Book] シンデレラで学ぶ平面幾何2004

    • Author(s)
      阿原一志
    • Total Pages
      175
    • Publisher
      シュプリンガーフェアラーク東京
    • Related Report
      2004 Annual Research Report
  • [Book] Energy of knots and conformal geometry2003

    • Author(s)
      J.O'Hara
    • Total Pages
      304
    • Publisher
      World Scientific
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] R.Langevin, J.O'HARA: "Conformally invariant energies of knots"J.Institut Math. Jussien. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] Jun O'HARA: "Energy of Knots and Conformal Geometry"World Scientific Publ.(Singapore). 304 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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