離散的な平均曲率一定曲面のクラスの広がり

• Project/Area Number: 19654010
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Kobe University
• Principal Investigator: ROSSMAN W.F (ROSSMAN Wayne F. / W.F Rossman) Kobe University, 理学研究科, 教授 (50284485)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,000,000 (Direct Cost: ¥3,000,000); Fiscal Year 2009: ¥900,000 (Direct Cost: ¥900,000); Fiscal Year 2008: ¥1,100,000 (Direct Cost: ¥1,100,000); Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000)
• Keywords: discrete differential geometry / discrete surfaces / Lie sphere geometry / hyperbolic 3-space / discrete surface / 離散的 / 平均曲率一定曲面 / special surface / 空間系 / 回転面 / Darboux変換

Research Abstract

今年度の前半に、Tim Hoffmann氏と佐々木武氏と吉田正章氏の協力を得て、双曲空間内の離散的な平坦曲面の定義を与えた。その定義に関して、事前に定義された双曲空間内の離散的な平均曲率1を持つ曲面との正しい関係があることを確認した。そのために、Galvez氏とMartinez氏とMilan氏の変分を用いた。その離散的な平坦曲面のcausticの定義も与えて、そのcausticの性質を調べた。
今年度の後半に、Fran Burstall氏とUdo Hertrich-Jeromin氏の協力を得て、リー球面幾何学のアプローで双曲空間内の平坦曲面を調べるために、曲面のクラスをOmega曲面のすべてに広げたほうが良いことがわかってきた。そうすると、接続の考え方が使え、またOmega曲面の集合の中から平坦曲面を選ぶために0位の保存量を用いた判定条件が得られた。平均曲率1の曲面を選ぶためにも、linear Weingarten surfaces of Bryant typeを選ぶためにも、それぞれの対応している保存量による判定条件があることを証明した。さらに、接続と保存量の考え方を用いて、その曲面を離散化する方法が明確になった。その離散化が前に定義された離散的な平均曲率1曲面と整合性があるかを確かめた。
また、神戸大学の院生の木ノ下祐輔氏の協力を得て、3次元Minkowski空間内の離散的な空間的平均曲率一定曲面の定義を見つけた。そして、ユークリッド空間内の離散的な平均曲率一定曲面について、1位の保存量によって定義されている離散的なガウス写像が、一般的には、離散的な調和写像にならないことを証明した。

Research Products

• [Journal Article] Holomorphic Representation of Constant Mean CurvatureSurfaces in Minkowski Space : Consequences of Non-Compactness in Loop Group Methods (2010)
  • Author(s): David Brander, Wayne Rossman, Nick Schmitt
  • Journal Title: Adv.Math. 223 Pages: 949-986
  • Peer Reviewed
• [Journal Article] Higher genus mean curvature 1 catenoids in hyperbolic and de Sitter 3-spaces (2010)
  • Author(s): Shoichi Fujimori, Wayne Rossman
  • Journal Title: Kyushu Math.J. (掲載確定)
  • Peer Reviewed
• [Journal Article] Isothermicity for discrete surfaces in the Euclidean and Minkowski 3-spaces (2010)
  • Author(s): Yusuke Kinoshita, Wayne Rossman
  • Journal Title: OCAMI Studies 3 Pages: 277-277
  • Peer Reviewed
• [Journal Article] 平均曲率一定曲面 (2010)
  • Author(s): W.Rossman
  • Journal Title: Iwanami Publication of Kyushu Math-for-Industry GCOE 25 Pages: 1-140
  • Peer Reviewed
• [Journal Article] Corresponding constant mean curvature surfaces in hyperbolic and Euclidean 3-spaces (2010)
  • Author(s): Wayne Rossman, Magdalena Toda
  • Journal Title: Pacific Journal of Applied Mathematics (掲載確定)
  • Peer Reviewed
• [Journal Article] Spacelike mean curvature one surfaces in de Sitter space (2009)
  • Author(s): S.Fujimori, W.Rossman, M.Umehara, K.Yamada, S-D.Yang
  • Journal Title: Comm.Anal.Geom. 17(掲載確定)
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of flat surfaces in hyperbolic 3-space (2009)
  • Author(s): M.Kokubu, W.Rossman, M.Umehara, K.Yamada
  • Journal Title: J.Math.Soc.Japan 61(3) Pages: 799-852
  • Peer Reviewed
• [Journal Article] New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter space (2009)
  • Author(s): S.Fujimori, W.Rossman, M.Umehara, K.Yamada, S-D.Yang
  • Journal Title: Res.Math. 56 Pages: 41-82
  • Peer Reviewed
• [Journal Article] Delaunay ends of constant mean curvature surfaces (2008)
  • Author(s): M. Kilian, W. Rossman &N. Schmitt
  • Journal Title: Compositio Math. 144 Pages: 186-220
  • Peer Reviewed
• [Journal Article] Period problems for mean curvature one surfaces in H^3 (2008)
  • Author(s): W. Rossman, M. Umehara, & K. Yamada
  • Journal Title: Advanced Studies in Pure Math. 51 Pages: 335-387
  • Peer Reviewed
• [Journal Article] Constant mean curvature surfaces in Euclidean and Minkowski three spaces (2008)
  • Author(s): D. Brander, w. Rossman & N. Schmitt
  • Journal Title: J. Geom. Symm. Phys. 12 Pages: 15-26
  • Peer Reviewed
• [Journal Article] Delaunay ends of constant mean curvature surfaces (2008)
  • Author(s): Rossman, Killian, Schmitt
  • Journal Title: Compositio Math. 144 Pages: 186-220
  • Peer Reviewed
• [Journal Article] Unitarization of monodromy representations and constant mean curva-ture trinoids in 3-dimensional space forms (2007)
  • Author(s): Rossman, Killian, Kobayashi, Schmitt
  • Journal Title: J. London Math. Soc. 75 Pages: 563-581
  • Peer Reviewed
• [Journal Article] Flat fronts in hyperbolic 3-space and their caustics (2007)
  • Author(s): Rossman, Kokubu, Umehara, Yamada
  • Journal Title: J. Math. Soc. Japan 59 Pages: 265-299
  • Peer Reviewed
• [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
• [Presentation] Discrete linear Weingarten surfaces in hyperbolic 3-space (2009)
  • Author(s): W.Rossman
  • Organizer: Geometry and something meeting
  • Place of Presentation: Fukuoka University(福岡県)
  • Year and Date: 2009-11-22
• [Presentation] Topological behavior of constant mean curvature surfaces (2009)
  • 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
• [Presentation] Discrete isothermic surfaces and applications to architecture (2009)
  • Author(s): W.Rossman
  • Organizer: Discrete Differential Geometry Miniworkshop
  • Place of Presentation: Technical University of Denmark(Denmark)
  • Year and Date: 2009-07-16
• [Presentation] A(limited)survey on the foundations of discrete differential geometry (2009)
  • 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
• [Presentation] Surfaces, DPW, and quantum cohomology (2009)
  • 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
• [Presentation] Basic Training for the DPW Method (2009)
  • Author(s): W.Rossman
  • Organizer: GEOSOCK joint geometry seminar
  • Place of Presentation: Osaka City University(大阪府)
  • Year and Date: 2009-05-13
• [Presentation] Discrete flat and linear Weingarten surfaces in hyperbolic 3-space (2009)
  • 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
• [Presentation] Discrete linear Weingarten surfaces in hyperbolic 3-space (2009)
  • Author(s): W.Rossman
  • Organizer: Differential Geometry with Mira
  • Place of Presentation: Osaka University(大阪府)
  • Year and Date: 2009-04-15
• [Presentation] Discrete surfaces and architecture (2008)
  • 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
• [Presentation] Discrete constant mean curvature surfaces via linear conserved quantities in any space form (2007)
  • Author(s): W. Rossman
  • Organizer: Progress in diff. geom. of submanifolds
  • Place of Presentation: Valenciennes, France
  • Year and Date: 2007-07-04