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A Weierstrass type representation for surfaces via loop group method and its applications

Research Project

Project/Area Number 26400059
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionHokkaido University

Principal Investigator

Kobayashi Shimpei  北海道大学, 理学研究院, 准教授 (40408654)

Project Period (FY) 2014-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords可積分曲面 / ループ群 / ワイエルシュトラス型の表現公式 / 停留曲面 / ソリトン方程式 / 極小曲面 / 曲線論 / 離散曲面 / ガウス曲率 / 離散化 / 曲面論 / 可積分系 / 調和写像
Outline of Final Research Achievements

Surfaces whose structure equation can be given by an integrable system are often called integrable surfaces. Here the integrable systems is a generic term used to refer to solvable (partial) differential equations. In particular many integrable surfaces have a Weierstrass type representation in terms of loop groups and holomorphic functions.
In this research we studied integrable surfaces by using the Weierstrass type representation. Concretely, we studied affine harmonic maps, constant Gaussian curvature surfaces in 3-dimensional hyperbolic space, discrete affine spheres, affine plane curves and maximal surfaces in 3-dimensional Anti-de Sitter space.

Report

(5 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (28 results)

All 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (3 results) Journal Article (5 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 5 results,  Acknowledgement Compliant: 4 results) Presentation (15 results) (of which Int'l Joint Research: 6 results,  Invited: 13 results) Book (1 results) Remarks (4 results)

  • [Int'l Joint Research] ミュンヘン工科大学(ドイツ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] 同済大学(中国)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Technical University of Munich(Germany)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Nonlinear d’Alembert formula for discrete pseudospherical surfaces2017

    • Author(s)
      Kobayashi Shimpei
    • Journal Title

      J. Geom. Phys.

      Volume: 119 Pages: 208-223

    • DOI

      10.1016/j.geomphys.2017.05.001

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A loop group method for minimal surfaces in the three-dimensional Heisenberg group2016

    • Author(s)
      Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi
    • Journal Title

      Asian Journal of Mathematics

      Volume: 20 Issue: 3 Pages: 409-448

    • DOI

      10.4310/ajm.2016.v20.n3.a2

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] A loop group method for affine harmonic maps into Lie groups2016

    • Author(s)
      Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi
    • Journal Title

      Advances in Mathematics

      Volume: 298 Pages: 207-253

    • DOI

      10.1016/j.aim.2016.04.018

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] A construction method for discrete constant negative Gaussian curvature surfaces2016

    • Author(s)
      Shimpei Kobayashi
    • Journal Title

      Mathematical Progress in Expressive Image Synthesis III, Extended and Selected Results from the Symposium MEIS2015

      Volume: 3 Pages: 21-33

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On the Bernstein Problem in the Three-dimensional Heisenberg Group2016

    • Author(s)
      Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi
    • Journal Title

      Canadian Mathematical Bulletin

      Volume: 59 Issue: 01 Pages: 50-61

    • DOI

      10.4153/cmb-2015-061-3

    • NAID

      120005756104

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Loop group method for surfaces with constant Gaussian curvature in the 3-dimensional hyperbolic space2018

    • Author(s)
      小林 真平
    • Organizer
      Geometry of Submanifolds and Integrable Systems
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 3次元ハイゼンベルグ群の極小曲面についてI, II2017

    • Author(s)
      小林 真平
    • Organizer
      幾何学阿蘇研究集会
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Minimal surfaces with symmetries in the three-dimensional Heisenberg group2017

    • Author(s)
      Shimpei Kobayashi
    • Organizer
      Higgs Bundles, Harmonic Maps, and Integrable Systems
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Surfaces with constant Gaussian curvature in the 3-dimensional hyperbolic space via loop groups2017

    • Author(s)
      Shimpei Kobayashi
    • Organizer
      Geometry colloquium
    • Place of Presentation
      マサチューセッツ大学(アメリカ合衆国・アマースト)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] A construction method for discrete indefinite affine spheres2017

    • Author(s)
      Shimpei Kobayashi
    • Organizer
      The Tenth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
    • Place of Presentation
      ジョージア大学(アメリカ合衆国・アセンズ)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Nonlinear d'Alembert formula for discrete pseudospherical surfaces2016

    • Author(s)
      Kobayashi Shimpei
    • Organizer
      OCAMI-KOBE-WASEDA Joint International Workshop on Differential Geometry and Integrable Systems
    • Place of Presentation
      神戸大学(兵庫県・神戸市)
    • Year and Date
      2016-02-16
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Constant negative Gaussian curvature surfaces in the 3-dimensional hyperbolic space2016

    • Author(s)
      Shimpei Kobayashi
    • Organizer
      One day workshop at Waseda Univeristy
    • Place of Presentation
      早稲田大学(東京都・新宿区)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 可積分系の手法によるリー群への調和写像2016

    • Author(s)
      Shimpei Kobayashi
    • Organizer
      第63回幾何学シンポジウム
    • Place of Presentation
      岡山大学(岡山県・岡山市)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 平均曲率一定曲面に対するワイエルシュトラス型の表現公式(入門)2016

    • Author(s)
      Shimpei Kobayashi
    • Organizer
      幾何学コロキウム
    • Place of Presentation
      北海道大学(北海道・札幌市)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] A construction method for discrete constant negative Gaussian curvature surfaces2015

    • Author(s)
      Kobayashi Shimpei
    • Organizer
      MEIS2015
    • Place of Presentation
      九州大学(福岡県・福岡市)
    • Year and Date
      2015-09-26
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 双曲平面への調和写像と曲面論への応用2014

    • Author(s)
      小林 真平
    • Organizer
      幾何学コロキウム
    • Place of Presentation
      東京大学(東京都・目黒区)
    • Year and Date
      2014-11-07
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 3次元ハイゼンベルグ群内の極小曲面について2014

    • Author(s)
      小林 真平
    • Organizer
      部分多様体とリー群作用2014
    • Place of Presentation
      東京理科大学(東京都・新宿区)
    • Year and Date
      2014-09-06
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] デモラン曲面のガウス写像による特徴付けについて2014

    • Author(s)
      小林 真平
    • Organizer
      関大微分幾何研究会
    • Place of Presentation
      関西大学(大阪府・吹田市)
    • Year and Date
      2014-07-06
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] A loop group method for affine harmonic maps into Lie groups2014

    • Author(s)
      Shimpei Kobayashi
    • Organizer
      RIMS研究集会 「群作用と部分多様体論の展開」
    • Place of Presentation
      京都大学数理解析研究所(京都府・京都市)
    • Year and Date
      2014-06-26
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Poisson's equations for inhomogeneous dielectric materials and neutral harmonic maps2014

    • Author(s)
      小林 真平
    • Organizer
      PDEセミナー
    • Place of Presentation
      北海道大学(北海道・札幌市)
    • Year and Date
      2014-04-17
    • Related Report
      2014 Research-status Report
    • Invited
  • [Book] 曲面とベクトル解析2016

    • Author(s)
      小林 真平
    • Total Pages
      216
    • Publisher
      日本評論社
    • Related Report
      2016 Research-status Report
  • [Remarks] Kobayashi Shimpei's Homepage

    • URL

      https://sites.google.com/site/kobayashishimpeisite/

    • Related Report
      2017 Annual Research Report
  • [Remarks]

    • URL

      https://sites.google.com/site/kobayashishimpeisite/

    • Related Report
      2016 Research-status Report
  • [Remarks] ホームページ

    • URL

      https://sites.google.com/site/kobayashishimpeisite/

    • Related Report
      2015 Research-status Report
  • [Remarks] ホームページ

    • URL

      https://sites.google.com/site/kobayashishimpeisite/research

    • Related Report
      2014 Research-status Report

URL: 

Published: 2014-04-04   Modified: 2019-03-29  

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