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Asymptotic behavior of surfaces with special properties via transformation theory and conserved quantities and integrable systems techniques

Research Project

Project/Area Number 20340012
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

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

Co-Investigator(Kenkyū-buntansha) NORO Masayuki  神戸大学, 大学院・理学研究科, 教授 (50332755)
KOIKE Tatsuya  神戸大学, 大学院・理学研究科, 准教授 (80324599)
小林 真平  弘前大学, 大学院・理工研究科, 助教 (40408654)
藤森 祥一  福岡教育大学, 教育学部, 准教授 (00452706)
國分 雅敏  東京電機大学, 工学部, 教授 (50287439)
Project Period (FY) 2008 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥17,420,000 (Direct Cost: ¥13,400,000、Indirect Cost: ¥4,020,000)
Fiscal Year 2011: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000)
Fiscal Year 2010: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2009: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2008: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Keywordssurface theory / asymptotics of surfaces / discrete differential geometry / linear Weingarten曲面 / Omega surfaces / 3-dimensional spaceforms / Moebius geometry / Lie sphere geometry / Lorentz spaceforms / conserved quantities / Riemannian connections / constant mean curvature surfaces / flat surfaces / linear Weingarten surfaces / 3-dimensional space forms / Weierstrass type representations / non-positive definite spaceforms / constant mean curvature / spacelike surfaces / Euclidean 3-space / hyperbolic 3-space / de Sitter 3-space / Minkowski 3-space / flat surface / spacelike surface
Research Abstract

The purpose of this research was to expand our understanding of discretizations in surface theory, in a way that would preserve the geometric mathematical structure (such as the notion of isothermicity, and various transformations like the Christoffel and Calapso and Darboux and Baecklund transformations) that exists for smooth surfaces.

Report

(6 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report   Self-evaluation Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (55 results)

All 2012 2011 2010 2009 2008 2007 Other

All Journal Article (30 results) (of which Peer Reviewed: 30 results) Presentation (23 results) (of which Invited: 4 results) Remarks (2 results)

  • [Journal Article] Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space2012

    • Author(s)
      T. Hoffmann, W. Rossman, T. Sasaki and M. Yoshida
    • Journal Title

      Trans. A.M.S.

      Volume: 364 Pages: 5605-5644

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lie geometry of linear Weingarten surfaces2012

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

      C. R. Acad. Sci. Paris, Ser. I

      Volume: 350 Pages: 413-416

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Corresponding constant mean curvature surfaces in hyperbolic and Euclidean 3-spaces2011

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

      Pacific Journal of Applied Mathematics

      Volume: 3 Pages: 37-42

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Triviality of the dressing isotropy for a Smyth-type potential and nonclosing of the resulting CMC surfaces2011

    • Author(s)
      J. F. Dorfmeister, W. Rossman
    • Journal Title

      Iasi conference proceedings

      Volume: 1 Pages: 61-70

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere2011

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

      Proc. Japan Acad.

      Volume: 87 Pages: 144-149

    • NAID

      40019127816

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lie geometry of flat fronts in hyperbolic space2010

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

      Comptes rendus-Mathematique Ser 1 348

      Pages: 661-664

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] The tt* structure of the quantum cohomology of CP1 from the viewpoint of differential geometry2010

    • Author(s)
      J. Dorfmeister, M.Guest, W.Rossman
    • Journal Title

      Asian J.Math. 14(3)

      Pages: 417-437

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Holomorphic representation of constant mean curvature surfaces in Minkowski space : consequences of noncompactness in loop group methods2010

    • Author(s)
      D.Brander, W.Rossman, N.Schmitt
    • Journal Title

      Adv.Math.

      Volume: 223 Pages: 949-986

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lie geometry of flat fronts in hyperbolic space2010

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

      Comptes Rendus, C.R.Acad.Sci.Paris

      Volume: 348 Pages: 661-664

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The tt^* structure of the quantum cohomology of CP1 from the viewpoint of differential geometry2010

    • Author(s)
      J.Dorfmeister, M.Guest, W.Rossman
    • Journal Title

      Asian J.Math.

      Volume: 14(3) Pages: 417-437

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Higher genus mean curvature 1 catenoids in hyperbolic and de Sitter 3-spaces2010

    • Author(s)
      S.Fujimori, W.Rossman
    • Journal Title

      Kyushu J.Math.

      Volume: 64 Pages: 169-180

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Discrete constant mean curvature surfaces via conserved quantities2010

    • Author(s)
      W.Rossman
    • Journal Title

      MI Lecture Note Series

      Volume: 25 Pages: 1-130

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Spacelike mean curvature 1 surfaces in de Sitter 3-space2009

    • Author(s)
      S. Fujimori, W. Rossman, M. Umehara, S-D. Yang, K. Yamada
    • Journal Title

      Comm. Anal. Geom.

      Volume: 17 Pages: 383-427

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Spacelike mean curvature 1 surfaces in de Sitter 3-space2009

    • Author(s)
      S.Fujimori, W.Rossman, M.Umehara, S-D.Yang, K.Yamada
    • Journal Title

      Comm.Anal.Geom. 17

      Pages: 383-427

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of flat surfaces in hyperbolic 3-space2009

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

      J.Math.Soc.Japan 61(3)

      Pages: 799-852

    • NAID

      10026998469

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter space2009

    • Author(s)
      S.Fujimori, W.Rossman, M.Umehara, K.Yamada, S-D.Yang
    • Journal Title

      Res.Math. 56

      Pages: 41-82

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Delaunay ends of constant mean curvature surfaces2008

    • Author(s)
      M. Kilian, W. Rossman, N. Schmitt
    • Journal Title

      Compositio Math.

      Volume: 144 Pages: 186-220

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Delaunay ends of constant mean curvature surfaces2008

    • Author(s)
      M.Kilian, W.Rossman, N.Schmitt
    • Journal Title

      Compositio Math. 144

      Pages: 186-220

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Delaunay ends of constant mean curvature surfaces2008

    • Author(s)
      M. Kilian, W. Rossman & N. Schmitt
    • Journal Title

      Compositio Math. 144

      Pages: 186-220

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Period problems for mean curvature one surfaces in H^32008

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

      Advanced Studies in Pure Math. 51

      Pages: 335-387

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Constant mean curvature surfaces in Euclidean and Minkowski three spaces2008

    • Author(s)
      D. Brander, W. Rossman & N. Schmitt
    • Journal Title

      J. Geom. Symm. Phys. 12

      Pages: 15-26

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lie geometry of flat fronts in hyperbolic space

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

      Comptes rendus - Mathematique(online publication complete: 10.1016/j.crma.2010.04.018).

      Volume: to appear

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Triviality of the dressing isotropy for a Smyth-type potential and nonclosing of the resulting CMC surfaces

    • Author(s)
      J.Dorfmeister, W.Rossman
    • Journal Title

      Iasi proceedings

      Volume: (To appear)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Corresponding constant mean curvature surfaces in hyperbolic and Euclidean 3-spaces

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

      Pacific J.Applied Math.

      Volume: (To appear)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Holomorphic Representation of Constant Mean CurvatureSurfaces in Minkowski Space : Consequences of Non-Compactness in Loop Group Methods

    • Author(s)
      David Brander, Wayne Rossman, Nick Schmitt
    • Journal Title

      Adv.Math. (To appear, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Higher genus mean curvature 1 catenoids in hyperbolic and de Sitter 3-spaces

    • Author(s)
      Shoichi Fujimori, Wayne Rossman
    • Journal Title

      Kyushu Math.J. (To appear, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Isothermicity for discrete surfaces in the Euclidean and Minkowski 3-spaces

    • Author(s)
      Yusuke Kinoshita, Wayne Rossman
    • Journal Title

      OCAMI Studies vol.3(To appear, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 平均曲率一定曲面

    • Author(s)
      W.Rossman
    • Journal Title

      Iwanami Publication of Kyushu Math-for-Industry GCOE (To appear, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Corresponding constant mean curvature surfaces in hyperbolic and Euclidean 3-spaces

    • Author(s)
      Wayne Rossman, Magdalena Toda
    • Journal Title

      Pacific Journal of Applied Mathematics (To appear, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Spacelike mean curvature one surfaces in de Sitter space

    • Author(s)
      S.Fujimori, W.Rossman, M.Umehara, K.Yamada, S-D.Yang
    • Journal Title

      Comm.Anal.Geom. (To appear, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] Global properties of surfaces, using various Weierstrass representations2012

    • Author(s)
      W. Rossman
    • Organizer
      geometry seminar
    • Place of Presentation
      Bath University, England
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Surfaces with Weierstrass representations and Lie sphere geometry2011

    • Organizer
      geometry seminar
    • Place of Presentation
      King's College, England.
    • Year and Date
      2011-02-10
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Surfaces with Weierstrass representations and Lie Sphere Geometry2011

    • Author(s)
      W.Rossman
    • Organizer
      幾何セミナー
    • Place of Presentation
      Kings College,(イギリス)(招待講演)
    • Year and Date
      2011-02-10
    • Related Report
      2010 Annual Research Report
  • [Presentation] Differential Geometric Methods for Discretizing Surface Theory2011

    • Author(s)
      W. Rossman
    • Organizer
      Geometry and Something 2011研究集会
    • Place of Presentation
      福岡大学
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Discrete linear Weingarten surfaces2011

    • Author(s)
      W. Rossman
    • Organizer
      談話会
    • Place of Presentation
      東京工業大学
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Discrete linear Weingarten surfaces2011

    • Author(s)
      W. Rossman
    • Organizer
      Workshop on Differential Geometry and Geometric Analysis
    • Place of Presentation
      東北大学
    • Related Report
      2011 Annual Research Report
    • Invited
  • [Presentation] Discrete Surfaces and Lie Sphere Geometry2010

    • Author(s)
      W.Rossman
    • Organizer
      Geometry Conference for China and Japan
    • Place of Presentation
      Northwest Uni.(中国)(招待講演)
    • Year and Date
      2010-09-07
    • Related Report
      2010 Annual Research Report
  • [Presentation] An application of Lie sphere geometry to surface theory in DDG2010

    • Author(s)
      W.Rossman
    • Organizer
      Workshop on Geometry & Integrable Systems
    • Place of Presentation
      東北大学(宮城県)(招待講演)
    • Year and Date
      2010-08-25
    • Related Report
      2010 Annual Research Report
  • [Presentation] リー球面幾何学とOmega曲面および曲面の微分幾何学的な離散化2010

    • Author(s)
      W.Rossman
    • Organizer
      幾何学セミナー
    • Place of Presentation
      東北大学(宮城県)(招待講演)
    • Year and Date
      2010-06-29
    • Related Report
      2010 Annual Research Report
  • [Presentation] Conserved quantities in surface theory2010

    • Organizer
      TIMS-OCAMI joint workshop on differential geometry
    • Place of Presentation
      National Taiwan University.
    • Year and Date
      2010-03-21
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] 離散的な微分幾何学と曲面理論(I)(II)(III)2010

    • Author(s)
      W.Rossman
    • Organizer
      離散可積分系・離散微分幾何学チュートリアル,九州大学産業技術数理研究センター 第9回ワークショップ
    • Place of Presentation
      Kyushu University(福岡県)
    • Year and Date
      2010-02-24
    • Related Report
      2009 Annual Research Report
  • [Presentation] Lie Sphere Geometry and Omega Surfaces2010

    • Author(s)
      W.Rossman
    • Organizer
      幾何学セミナー
    • Place of Presentation
      Bath Uni.(イギリス)(招待講演)
    • Related Report
      2010 Annual Research Report
  • [Presentation] Discrete linear Weingarten surfaces in hyperbolic 3-space2009

    • Author(s)
      W.Rossman
    • Organizer
      Geometry and something meeting
    • Place of Presentation
      Fukuoka University(福岡県)
    • Year and Date
      2009-11-22
    • Related Report
      2009 Annual Research Report
  • [Presentation] Topological behavior of constant mean curvature surfaces2009

    • Author(s)
      W.Rossman
    • Organizer
      The First KOOK-TAPU Joint Seminar
    • Place of Presentation
      Osaka City University(大阪府)
    • Year and Date
      2009-08-20
    • Related Report
      2009 Annual Research Report
  • [Presentation] Discrete isothermic surfaces, and applications to architecture2009

    • Organizer
      Discrete differential geometry miniworkshop
    • Place of Presentation
      Technical University of Denmark.
    • Year and Date
      2009-07-16
    • Related Report
      2011 Final Research Report 2010 Self-evaluation Report
  • [Presentation] Discrete isothermic surfaces and applications to architecture2009

    • Author(s)
      W.Rossman
    • Organizer
      Discrete Differential Geometry Miniworkshop
    • Place of Presentation
      Technical University of Denmark(Denmark)
    • Year and Date
      2009-07-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] A(limited)survey on the foundations of discrete differential geometry2009

    • Author(s)
      W.Rossman
    • Organizer
      Variational Problems for Curves and Surfaces and Related Topics「曲線と曲面の変分問題と関連する話題」
    • Place of Presentation
      Nara Women's University(奈良県)
    • Year and Date
      2009-07-01
    • Related Report
      2009 Annual Research Report
  • [Presentation] Surfaces, DPW, and quantum cohomology2009

    • Author(s)
      W.Rossman
    • Organizer
      New aspects of surface theory
    • Place of Presentation
      Shuto University(TMU)(東京都)
    • Year and Date
      2009-05-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] Basic Training for the DPW Method2009

    • Author(s)
      W.Rossman
    • Organizer
      GEOSOCK joint geometry seminar
    • Place of Presentation
      Osaka City University(大阪府)
    • Year and Date
      2009-05-13
    • Related Report
      2009 Annual Research Report
  • [Presentation] Discrete flat and linear Weingarten surfaces in hyperbolic 3-space2009

    • Author(s)
      W.Rossman
    • Organizer
      Fusion of Integrable Systems and Geometry
    • Place of Presentation
      Tohoku University(宮城県)
    • Year and Date
      2009-04-18
    • Related Report
      2009 Annual Research Report
  • [Presentation] Discrete linear Weingarten surfaces in hyperbolic 3-space2009

    • Author(s)
      W.Rossman
    • Organizer
      Differential Geometry with Mira
    • Place of Presentation
      Osaka University(大阪府)
    • Year and Date
      2009-04-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] Discrete surfaces and architecture2008

    • Author(s)
      W. Rossman
    • Organizer
      国際研究集会「リーマン面、調和写像と可視化」
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2008-12-15
    • Related Report
      2008 Annual Research Report
  • [Presentation] Discrete constant mean curvature surfaces via linear conserved quantities in any space form2007

    • Organizer
      Progress in surface theory meeting
    • Place of Presentation
      Oberwolfach Institute, Germany.
    • Year and Date
      2007-05-01
    • Related Report
      2011 Final Research Report
  • [Remarks]

    • URL

      http://math.kobe-u.ac.jp/home-j/wayne.html

    • Related Report
      2011 Final Research Report
  • [Remarks] ホームページ等

    • URL

      http://math.kobe-u.ac.jp/home-j/wayne.html

    • Related Report
      2010 Self-evaluation Report

URL: 

Published: 2008-04-01   Modified: 2016-04-21  

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