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離散化された曲面の微分幾何学的様相

Research Project

Project/Area Number 15K04845
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

Rossman W.F  神戸大学, 理学研究科, 教授 (50284485)

Co-Investigator(Kenkyū-buntansha) 直川 耕祐  神戸大学, 理学研究科, 特別研究員(PD) (60740826) [Withdrawn]
佐治 健太郎  神戸大学, 理学研究科, 准教授 (70451432)
Project Period (FY) 2015-04-01 – 2020-03-31
Project Status Granted (Fiscal Year 2018)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywordssurface theory / discrete geometry / minimal surface / Omega surface / transformation theory / Lie sphere geometry / channel surfaces / orthogonal systems / Guichard surfaces / Omega surfaces / discrete surfaces / semi-discrete surfaces / integrable systems / Moebius geometry / 離散的微分幾何学 / 離散曲面 / 半離散曲面
Outline of Annual Research Achievements

微分幾何の観点から、離散化された曲面の理論を新たに構築し、曲面の構成法を新たに与え、現れる性質の解析を行った。具体的に得られた性質は以下の通りである。
(1) Francis Burstall氏(バース大学)、Udo Hertrich-Jeromin氏(ウィーン工科大学)との共同研究において、オメガ曲面と呼ばれる曲面の離散化の理論を開発した。これにより、曲面の離散化の研究対象が大幅に拡がり、具体例の構成や性質解析等に向けた研究が行われることが期待される。
(2) 安本真士氏(大阪市立大学)との共同研究において、ワイエルストラス型の表現公式を持つ半離散線形ワインガルテン曲面の構成法を新たに導出し、現れる特異点の解析を行った。これにより、半離散曲面の構造解析という新たな研究テーマを開拓した。
(3) Seong-Deog Yang氏(高麗大学校)、Joseph Cho氏(神戸大学博士課程学生)との共同研究で、離散極小曲面が対称性を持つための条件を発見し、新たな離散極小曲面の具体例を構成した。
(4) 藤森祥一氏(岡山大学)、川上裕氏(金沢大学)、國分雅敏氏(東京電機大学)、梅原雅顕氏(東京工業大学)、山田光太郎氏(東京工業大学)との共同研究で、ローレンツ軽量を持つ空間系内の平均曲率一定曲面の解析的延長の性質を調べて、特にド・ジッター空間内の平均曲率一定曲面を調べた。

Current Status of Research Progress
Current Status of Research Progress

1: Research has progressed more than it was originally planned.

Reason

(1) 共同研究者のFrancis Burstall氏とUdo Hertrich-Jeromin氏との活発な研究打ち合わせによって、離散オメガ曲面の理論の基礎理論の構築を完成させることに成功した。これは当初の想定よりも早いものである。
(2) 安本真士氏との共同研究により、単に半離散等温曲面とは限らない半離散曲面を構成するだけでなく、半離散曲面の性質を解析するという新たな研究の開拓にも成功している。より一般の離散化された曲面の性質の解析は、今後の研究の新たな方向性を与えるものである。
(3) 上記の(2)と関連して、Seong-Deog Yang氏とJoseph Cho氏との共同研究は、離散曲面の大域的性質の解析に成功したものである。この成果は当初想定していなかったものであり、より一層の研究の発展に寄与するものである。
(4) 藤森祥一氏、川上裕氏、國分雅敏氏、梅原雅顕氏、山田光太郎氏との共同研究は、ローレンツ軽量を持つ空間系内の平均曲率一定曲面の解析的延長に関する結果を得た。

Strategy for Future Research Activity

これまでの結果を踏まえて、次年度は以下の問題に取り組む。
(1) Seong-Deog Yang氏と安本真士氏とJoseph Cho氏と共同で、離散化された平均曲率一定曲面の大域的性質をさらに調べる。特に、これらの曲面が対称性を持つための条件を詳しく調べ、新たな例を構築する手法を確立する。
(2) Francis Burstall氏とUdo Hertrich-Jeromin氏との共同研究をさらに発展させて、離散オメガ曲面の変換理論を新たに整備するとともに、離散曲面とは限らないより一般のcurved flatの離散化に着手する。
(3) 不定値計量を持つ空間内のなめらかな平均曲率一定曲面の解析的延長を研究する。

Report

(4 results)
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report

Research Products

(43 results)

All 2019 2018 2017 2016 2015 Other

All Int'l Joint Research Journal Article Presentation Book Funded Workshop

  • [Int'l Joint Research] Technical Univ. Vienna(オーストリア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Bath Univ.(英国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Technical Univ. Munich(ドイツ)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Polynomial Conserved Quantities of Lie Applicable Surfaces2019

    • Author(s)
      W. Rossman, U. Hertrich-Jeromin, F. Burstall, M. Pember
    • Journal Title

      Manuscripta Math.

      Volume: 158 Pages: 505-546

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Addendum: singularities of flat fronts in hyperbolic space2018

    • Author(s)
      M. Kokubu, W. Rossman, K. Saji, M. Umehara, K. Yamada
    • Journal Title

      Pacific J. Math

      Volume: 294 Pages: 505-509

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Discrete linear Weingarten surfaces2018

    • Author(s)
      F. Burstall, U. Hertrich-Jeromin, W. Rossman
    • Journal Title

      Nagoya Math. J.

      Volume: 231 Pages: 55-58

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Discrete linear Weingarten surfaces with singularities in Riemannian and Lorentzian spaceforms2018

    • Author(s)
      W. Rossman, M. Yasumoto
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 78 Pages: 383-410

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Analytic extension of exceptional constant mean curvature one catenoids in de Sitter 3-space2018

    • Author(s)
      S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, K. Yamada
    • Journal Title

      Math. J. Okayama Univ.

      Volume: to appear

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Characterizing singularities of a surface in Lie sphere geometry2018

    • Author(s)
      M. Pember, W. Rossman, K. Saji, K. Teramoto
    • Journal Title

      Hokkaido Math. J

      Volume: to appear

    • Related Report
      2018 Research-status Report
  • [Journal Article] Semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities2018

    • Author(s)
      W. Rossman, M. Yasumoto
    • Journal Title

      Osaka Journal of Mathematics

      Volume: to appear

    • Related Report
      2018 Research-status Report
  • [Journal Article] Discrete channel surfaces2018

    • Author(s)
      U. Hertrich-Jeromin, W. Rossman, G. Szewieczek
    • Journal Title

      Mathematische Zeitschrift

      Volume: to appear

    • Related Report
      2018 Research-status Report
  • [Journal Article] Discrete linear Weingarten surfaces2018

    • Author(s)
      Burstall, F., Hertrich-Jeromin, U. and Rossman, W.
    • Journal Title

      Nagoya Mathematical Journal, to appear

      Volume: --

    • DOI

      10.1017/nmj.2017.11

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Addendum: singularities of flat fronts in hyperbolic space2018

    • Author(s)
      Kokubu, M., Rossman, W., Saji, K., Umehara, M. and Yamada, K.
    • Journal Title

      Pacific J. Math., to appear

      Volume: --

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Polynomial conserved quantities of Lie applicable surfaces2018

    • Author(s)
      Burstall, F., Hertrich-Jeromin, U., Pember, M. and Rossman, W.
    • Journal Title

      Manuscripta Math., to appear

      Volume: --

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Analytic extension of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space2017

    • Author(s)
      Fujimori, S., Kawakami, Y., Kokubu, M., Rossman, W., Umehara, M. and Yamada, K.
    • Journal Title

      Osaka J. Math.

      Volume: 54 Pages: 249-272

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Magnetic geodesics on surfaces with singularities2017

    • Author(s)
      Branding, V., Rossman, W.
    • Journal Title

      Pacific Journal of Mathematics for Industry

      Volume: 9:3 Pages: 29-36

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Discrete linear Weingarten surfaces2017

    • Author(s)
      F. Burstall, U. Hertrich-Jeromin and W. Rossman
    • Journal Title

      to appear in Nagoya Math J.

      Volume: -

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Entire zero-mean curvature graphs of mixed type in Lorentz-Minkowski 3-space2017

    • Author(s)
      S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara and K. Yamada
    • Journal Title

      to appear in Quart. J. Math.

      Volume: -

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Analytic extension of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space2017

    • Author(s)
      S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara and K. Yamada
    • Journal Title

      to appear in Osaka Journal of Mathematics

      Volume: -

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Magnetic geodesics on surfaces with singularities2017

    • Author(s)
      V. Branding and W. Rossman
    • Journal Title

      to appear in Pacific Journal of Mathematics for Industry

      Volume: -

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Discrete linear Weingarten surfaces and their singularities in Riemannian and Lorentzian spaceforms2017

    • Author(s)
      W. Rossman and M. Yasumoto
    • Journal Title

      to appear in Advanced Studies in Pure Mathematics

      Volume: -

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Semi-discrete isothermic surfaces2016

    • Author(s)
      F. Burstall, U. Hertrich-Jeromin, C. Muller and W. Rossman
    • Journal Title

      Geom. Dedicata

      Volume: 183 Pages: 43-58

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Erratum to the article "Zero mean curvature surfaces in Lorentz-Minkowski 3-space which change type across a light-like line''2016

    • Author(s)
      S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara, K. Yamada and S.-D. Yang
    • Journal Title

      Osaka J. Math.

      Volume: 53 Pages: 289-292

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Semi-discrete isothermic surfaces2016

    • Author(s)
      F. Burstall, U. Hertrich-Jeromin, C. Mueller and W. Rossman
    • Journal Title

      Geometriae dedicata

      Volume: 印刷中

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Zero mean curvature surfaces in Lorentz-Minkowski 3-space and 2-dimensional fluid mechanics2015

    • Author(s)
      Shoichi Fujimori, Young Wook Kim, Sung-Eun Koh, Wayne Rossman, Heayong Shin, Masaaki Umehara, Kotaro Yamada, Seong-Deog Yang
    • Journal Title

      Mathematical Journal of Okayama University

      Volume: 57 Pages: 173-200

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Zero mean curvature surfaces in Lorentz-Minkowski 3-space which change type across a light-like line2015

    • Author(s)
      Shoichi Fujimori, Young Wook Kim, Sung-Eun Koh, Wayne Rossman, Heayong Shin, Masaaki Umehara, Kotaro Yamada, Seong-Deog Yang
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 52 Pages: 285-297

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Discrete special isothermic surfaces2015

    • Author(s)
      F. Burstall, U. Hertrich-Jeromin, W. Rossman and S. Santos
    • Journal Title

      Geometriae dedicata

      Volume: 174 Pages: 1-11

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Surfaces in Moebius (and Lie sphere) geometry2018

    • Author(s)
      W. Rossman
    • Organizer
      Joint Japan-Korea Workshop on DDG, Korea University, Seoul
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Surface discretization and Lie sphere geometry2018

    • Author(s)
      W. Rossman
    • Organizer
      SIDE 13 Conference, Kyushu University
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Introduction to Lie sphere geometry and applications to surface theory2018

    • Author(s)
      W. Rossman
    • Organizer
      Summer School 2018 in Fukuoka, "Geometric shape generation", Kyushu University
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Discrete omega surfaces2018

    • Author(s)
      W. Rossman
    • Organizer
      Mini-Workshop on Geometry of Surfaces and Related Topics, Kobe University
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Singularities on discrete omega surfaces2018

    • Author(s)
      W. Rossman
    • Organizer
      Minimal Surfaces: Integrable Systems and Visualization, Technical University Munich
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Discrete Omega surfaces and their singularities2018

    • Author(s)
      W. Rossman
    • Organizer
      Applications of Differential Geometry, Integrability, Singularity Theory, Hiroshima Institute of Technology
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Singularities of semi-discrete linear Weingarten surfaces2017

    • Author(s)
      Rossman, W.
    • Organizer
      大阪市立大学の微分幾何学セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Surface discretizations that preserve curvature characteristics2016

    • Author(s)
      W. Rossman
    • Organizer
      第63回幾何学シンポジウム
    • Place of Presentation
      岡山大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Using integrable systems techniques to discretize surface theory2016

    • Author(s)
      W. Rossman
    • Organizer
      Oktoberfest geometry workshop
    • Place of Presentation
      早稲田大学
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] From discrete surface theory to architecture and back2016

    • Author(s)
      W. Rossman
    • Organizer
      Geometry and Material Sciences (GEMS) Workshop
    • Place of Presentation
      Okinawa Institute of Science and Technology
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Discrete linear Weingarten surfaces: singularities of discrete surfaces2015

    • Author(s)
      W. Rossman
    • Organizer
      福岡大学微分幾何研究会
    • Place of Presentation
      福岡大学(福岡県福岡市)
    • Year and Date
      2015-10-30
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Topology and computer graphics as aids in mathematical surface theory, with applications2015

    • Author(s)
      W. Rossman
    • Organizer
      Kobe Studio Computer Graphics Seminar
    • Place of Presentation
      神戸大学(兵庫県神戸市)
    • Year and Date
      2015-06-26
    • Related Report
      2015 Research-status Report
    • Invited
  • [Book] Differential Geometry of Curves and Surfaces2017

    • Author(s)
      Umehara, M., Yamada, K., Translator: Rossman, W.
    • Total Pages
      312
    • Publisher
      World Scientific Publishing Co.
    • ISBN
      9789814740234
    • Related Report
      2017 Research-status Report
  • [Book] 英語翻訳:Differential Geometry of Curves and Surfaces2017

    • Author(s)
      M. Umehara, K. Yamada (Syokabo). 翻訳者:Wayne Rossman,
    • Publisher
      to appear with World Scientific Publishing Co.
    • Related Report
      2016 Research-status Report
  • [Funded Workshop] The 2nd International Conference, Geometry of Submanifolds and Integrable Systems, Osaka City University2019

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] Geometry of Submanifolds and Integrable Systems, Osaka City University2018

    • Related Report
      2018 Research-status Report

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Published: 2015-04-16   Modified: 2019-12-27  

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