Global construction of constant mean curvature surfaces in terms of contact geometry and loop groups

• Project/Area Number: 21540067
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Yamagata University
• Principal Investigator: INOGUCHI Jun-ichi 山形大学, 理学部, 准教授 (40309886)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 平均曲率一定曲面 / 接触幾何学 / ループ群 / 調和写像 / DPW法 / 極小曲面 / 差分平面曲線 / τ関数 / 移動曲面 / ハイゼンベルグ群 / 原始写像 / ツイスター構成法 / 平面曲線 / 離散曲線 / 単位接ベクトル束 / ツイスター空間 / f構造

Research Abstract

We showed that minimal surfaces in hyperbolic 3-space are obtained as projections of f-holomorphic curves in the semi-Riemannian homogeneous contact space SL (2,C)/U(1). By using the appropriate loop group splitting, for any prescribed potential, we can construct f-holomorphic curves in SL (2,C)/U(1). It is shown that non-minimal constant mean curvature surfaces with mean curvature less than 1 can be obtained from f-holomorphic curves. By using this loop group method (new DPW-method), we constructed radially symmetric constant mean curvature surfaces in hyperbolic 3-space. We classified minimal translation surfaces in the 3-dimensional Heisenberg group.

Research Products

• [Journal Article] Constant mean curvature surfaces in hyperbolic 3-spacevia loop groups (2014)
  • Author(s): J.F.Dorfmeister, J.Inoguchi, S.P.Kobayashi
  • Journal Title: Journal die fur reine und angewandte Mathematik Volume: (印刷中) Issue: 686 Pages: 1-36
  • DOI: 10.1515/crelle-2012-0024
  • Peer Reviewed
• [Journal Article] Real hypersurfaces in complex space forms with pseudo-parallel Ricci operator (2012)
  • Author(s): J. Inoguchi
  • Journal Title: Geometry-Dynamical Systems (Electronic) Volume: 14 Pages: 69-89
  • Peer Reviewed
• [Journal Article] Submanifolds with harmonic mean curvature in pseudo-Hermitian geometry (2012)
  • Author(s): J. Inoguchi, J. E. Lee
  • Journal Title: Archivum Mathematicum Volume: Brnno 48 Pages: 15-26
  • Peer Reviewed
• [Journal Article] Explicit solutions to semi-discrete modified KdV equation and motion of discrete plane curves (2012)
  • Author(s): J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta
  • Journal Title: Journal of Physics A Volume: 45
  • Peer Reviewed
• [Journal Article] Submanifolds with harmonic mean curvature in pseudo-Hermitian geometry (2012)
  • Author(s): J.Inoguchi, J.E.Lee
  • Journal Title: Archivum Mathematicum, Brno Volume: 48 Issue: 1 Pages: 15-26
  • DOI: 10.5817/am2012-1-15
  • Peer Reviewed
• [Journal Article] Real hypersurfaces in complex space forms with pseudo-parallel Ricciopeartor (2012)
  • Author(s): J.Inoguchi
  • Journal Title: Geometry-Dynamical Systems (Electronic) Volume: 14 Pages: 69-89
  • Peer Reviewed
• [Journal Article] Minimal translation surfaces in the Heisenberg group Nil_3 (2012)
  • Author(s): J.Inoguchi L.Lopez, M.I.Munteanu
  • Journal Title: Geometriae Dedicata Volume: (印刷中) Issue: 1 Pages: 221-231
  • DOI: 10.1007/s10711-012-9702-8
  • Peer Reviewed
• [Journal Article] Discrete integrable systems and hodograph transformation arising from motions of discrete plane curves (2011)
  • Author(s): B.-F. Feng, J. Inoguchi, K. Kajiwara, M. Maruno
  • Journal Title: Journal of Physics A Volume: 44
  • Peer Reviewed
• [Journal Article] Grassmann geometry on the 3-dimensional unimodular Lie groups II (2011)
  • Author(s): J. Inoguchi, H. Naitoh
  • Journal Title: Hokkaido Mathematical Journal Volume: 40 Pages: 411-429
  • Peer Reviewed
• [Journal Article] Curvatures and symmetries of tangent sphere bundles (2011)
  • Author(s): J. T. Cho, J. Inoguchi
  • Journal Title: Houston Journal of Mathematics Volume: 37 Pages: 1125-1142
  • Peer Reviewed
• [Journal Article] Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves (2011)
  • Author(s): B.-F.Feng, J.Inoguchi, K.Kajiwara, K.Maruno and Y.Ohta
  • Journal Title: J. Phys. A : Math. Theor. Volume: 44 Issue: 39 Pages: 19-19
  • DOI: 10.1088/1751-8113/44/39/395201
  • Peer Reviewed
• [Journal Article] Classsification and isolation phenomena of biharmonic maps and bi-Yang-Mills fields (2010)
  • Author(s): T. Ichiyama, J. Inoguchi, H. Urakawa
  • Journal Title: Note di Matematica Volume: 30 Pages: 15-48
  • Peer Reviewed
• [Journal Article] Ruled real hypersurfaces in complex space forms (2010)
  • Author(s): T. Hamada, J. Inoguchi
  • Journal Title: Kodai Mathematical Journal Volume: 33 Pages: 123-134
  • NAID: 130004687926
  • Peer Reviewed
• [Journal Article] On three-dimensional real hypersurfaces in complex space forms (2010)
  • Author(s): J. T. Cho, T. Hamada, J. Inoguchi
  • Journal Title: okyo Journal of Mathematics Volume: 33 Pages: 31-47
  • Peer Reviewed
• [Journal Article] とり扱いの難しい可積分方程式 (2010)
  • Author(s): 井ノ口順一
  • Journal Title: 九州大学応用力学研究所研究報告21ME-S2 Pages: 7-15
• [Journal Article] On three-dimensional real hypersurfaces in complex space forms (2010)
  • Author(s): J.T.Cho, T.Hamada, J.Inoguchi
  • Journal Title: Tokyo Journal of Mathematics Volume: 33 Pages: 31-47
  • Peer Reviewed
• [Journal Article] Ruled real hypersurfaces in complex space forms (2010)
  • Author(s): Hamada, 井ノ口順一
  • Journal Title: Kodai Mathematical Journal 33 Pages: 123-134
  • NAID: 130004687926
  • Peer Reviewed
• [Journal Article] とり扱いの難しい可積分方程式 (2010)
  • Author(s): 井ノ口順一
  • Journal Title: 九州大学応用力学研究所研究報告 21ME-S2 Pages: 7-15
• [Journal Article] Constant mean curvature surfaces in hyperbolic 3-space via loop groups
  • Author(s): J. F. Dorfmeister, J. Inoguchi, S. P. Kobayashi
  • Journal Title: Journal die fur reine und angewandte Mathematik Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] Affine biharmonic biharmonic curves in 3-dimensional homogeneous geometries
  • Author(s): J. Inoguchi, J.-E. Lee
  • Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] Motion and Backlund transformations of plane discrete curves
  • Author(s): J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta
  • Journal Title: Kyushu Journal of Mathematics Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] Minimal translation surfaces in the Heisenberg group Nil_3
  • Author(s): J. Inoguchi, L. Lopez, M. I. Munteanu
  • Journal Title: Geometriae Dedicata Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] Affine biharmonic curves in 3-dimensional homogeneous geometries
  • Author(s): J.Inoguchi, J.-E.Lee
  • Journal Title: Mediterranean Journal of Mathematics Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] Motion and Backlund transformations of plane discrete curves
  • Author(s): J.Inoguchi, K.Kajiwara, N.Matsuura, Y.Ohta
  • Journal Title: Kyushu Journal of Mathematics Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] Classsification and isolation phenomena of biharmonic maps and bi-Yang-Mills fields
  • Author(s): T.Ichiyama., 井ノ口順一, H.Urakawa
  • Journal Title: Note di Matematica (掲載決定)
  • Peer Reviewed
• [Presentation] 差分幾何 (2011)
  • Author(s): 井ノ口順一
  • Organizer: 日本数学会幾何学分科会第58回幾何学シンポジウム
  • Place of Presentation: 山口大学
  • Year and Date: 2011-08-30
• [Presentation] 差分幾何 (2011)
  • Author(s): 井ノ口順一
  • Organizer: 第58回幾何学シンポジウム
  • Place of Presentation: 山口大学(招待講演)
  • Year and Date: 2011-08-30
• [Presentation] 曲面の差分幾何 (2011)
  • Author(s): 井ノ口順一
  • Organizer: RIMS研究集会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-08-17
• [Presentation] 曲面の差分幾何 (2011)
  • Author(s): 井ノ口順一
  • Organizer: RIMS研究集会,可積分系数理の進化
  • Place of Presentation: 京都大学数理解析研究所(招待講演)
  • Year and Date: 2011-08-17
• [Presentation] Constant mean curvature surfaces in hyperbolic 3-space (2011)
  • Author(s): 井ノ口順一
  • Organizer: 日本・スペイン微分幾何学研究集会(特別講演)
  • Place of Presentation: スペイン・グラナダ大学
• [Presentation] Constant mean curvature surfaces in hyperbolic 3-space (2011)
  • Author(s): 井ノ口順一
  • Organizer: 日本・スペイン微分幾何学研究集会特別講演
  • Place of Presentation: スペイングラナダ大学(招待講演)
• [Presentation] 3次元等質空間の微分幾何 (2010)
  • Author(s): 井ノ口順一
  • Organizer: 広島幾何学研究集会(特別講演)
  • Place of Presentation: 広島大学
  • Year and Date: 2010-10-08
• [Presentation] 3次元等質空間の微分幾何1, 2 (2010)
  • Author(s): 井ノ口順一
  • Organizer: 広島幾何学研究集会特別講演
  • Place of Presentation: 広島大学(招待講演)
  • Year and Date: 2010-10-08
• [Presentation] 3次元双曲空間の平均曲率一定曲面 (2010)
  • Author(s): 井ノ口順一
  • Organizer: 日本数学会秋季総合分科会幾何学分科会特別講演
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-23
• [Presentation] 3次元双曲空間の平均曲率一定曲面 (2010)
  • Author(s): 井ノ口順一
  • Organizer: 日本数学会秋季総合分科会幾何学分科会特別講演
  • Place of Presentation: 名古屋大学(招待講演)
  • Year and Date: 2010-09-23
• [Presentation] Twistorial construction of hyperbolic minimal surfaces (2010)
  • Author(s): 井ノ口順一
  • Organizer: 国際会議「第6回中日友好幾何学研究集会」
  • Place of Presentation: 中国・西北大学
  • Year and Date: 2010-09-05
• [Presentation] Twistorial construction of hyperbolic minimal surfaces (2010)
  • Author(s): 井ノ口順一
  • Organizer: 国際会議「第6回中日友好幾何学研究集会」
  • Place of Presentation: 中国西北大学(招待講演)
  • Year and Date: 2010-09-05
• [Presentation] 3次元双曲空間の平均曲率一定曲面 (2009)
  • Author(s): 井ノ口順一
  • Organizer: 日本数学会幾何学分科会第56回幾何シンポジウム
  • Place of Presentation: 佐賀大学
  • Year and Date: 2009-08-30
• [Presentation] 3次元双曲空間の平均曲率一定曲面 (2009)
  • Author(s): 井ノ口順一
  • Organizer: 日本数学会幾何学分科会 第56回幾何シンポジウム
  • Place of Presentation: 佐賀大学
  • Year and Date: 2009-08-30
• [Book] 離散可積分系・離散微分幾何チュートリアル (2012)
  • Author(s): 井ノ口順一, 太田泰広, 筧三郎, 梶原健司, 松浦望
  • Publisher: 九州大学
• [Book] 離散可積分系・離散微分幾何チュートリアル2012(分担執筆) (2012)
  • Author(s): 井ノ口順一
  • Publisher: 九州大学COEレクチャーノート40
• [Book] リッカチのひ・み・つ (2010)
  • Author(s): 井ノ口順一
  • Total Pages: 256
  • Publisher: 日本評論社
• [Book] 曲線とソリトン (2010)
  • Author(s): 井ノ口順一
  • Total Pages: 192
  • Publisher: 朝倉書店