Fusion of discrete and smooth integrable geometry

• Project/Area Number: 18K03265
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Hokkaido University
• Principal Investigator: Kobayashi Shimpei 北海道大学, 理学研究院, 准教授 (40408654)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 調和写像 / 可積分系 / 離散曲面 / ガウス写像 / 離散平均曲率一定曲面 / 極小曲面 / ハイゼンベルグ群 / 統計多様体 / 離散微分幾何 / 時間的極小曲面 / アフィン平面曲線 / ハイゼンベルグ / 時間的曲面 / ループ群 / 多次元正規分布 / 特異点 / Ruh-Vilmsの定理 / 極小ラグランジアン曲面 / Demoulin曲面 / 甘利-Chentsovのアルファ接続 / ワイエルシュトラス型の表現公式 / アフィン・カッツ-ムーディ代数 / 可積分幾何 / 曲面論 / 微分幾何

Outline of Final Research Achievements

I studied on stationary surfaces and harmonic maps formulated as variational problems. Specifically, I comprehensively studied integrable surfaces with infinite-dimensional symmetries and formulated their discretization. My research significantly extended the conventional approach to discretizing constant mean curvature surfaces and formulated it on a general graph. I also conducted a comprehensive study of various integrable surfaces, such as minimal Lagrangian surfaces, minimal surfaces in the three-dimensional Heisenberg groups, and Demoulin surfaces, through harmonicities of Gauss maps.

Academic Significance and Societal Importance of the Research Achievements

変分問題として定式化される停留曲面は，シャボン玉をはじめ現実世界によく見られる重要な曲面である．本研究は，停留曲面のうち，いわゆる無限次元の対称性をもつ可積分曲面の微分幾何学的研究であり，その離散化についても研究した．離散的な曲面は，近年工学や建築学などの「ものづくり」で非常に注目される対象であり，本研究では，平均曲率一定曲面（体積一定の条件の下，面積汎関数の停留曲面である）の離散化を従来より広い枠組みで定式化した．今後，この定式化を用いてさまざまな分野への応用が期待できる．

Research Products

• [Int'l Joint Research] ミュンヘン工科大学(ドイツ)
• [Int'l Joint Research] デンマーク工科大学(デンマーク)
• [Int'l Joint Research] ミュンヘン工科大学(ドイツ)
• [Int'l Joint Research] デンマーク工科大学(デンマーク)
• [Int'l Joint Research] グルノーブル大学(フランス)
• [Int'l Joint Research] ミュンヘン工科大学(ドイツ)
• [Int'l Joint Research] デンマーク工科大学(デンマーク)
• [Int'l Joint Research] ミュンヘン工科大学(ドイツ)
• [Int'l Joint Research] 清華大学(中国)
• [Int'l Joint Research] ミュンヘン工科大学(ドイツ)
• [Journal Article] On a constant curvature statistical manifold (2022)
  • Author(s): Kobayashi Shimpei、Ohno Yu
  • Journal Title: Information Geometry Volume: 5 Issue: 1 Pages: 31-46
  • DOI: 10.1007/s41884-022-00065-x
  • Peer Reviewed
• [Journal Article] Timelike Minimal Surfaces in the Three-Dimensional Heisenberg Group (2022)
  • Author(s): Kiyohara Hirotaka、Kobayashi Shimpei
  • Journal Title: The Journal of Geometric Analysis Volume: 32 Issue: 8 Pages: 225-225
  • DOI: 10.1007/s12220-022-00962-8
  • Peer Reviewed / Open Access
• [Journal Article] Discrete constant mean curvature surfaces on general graphs (2022)
  • Author(s): Hoffmann Tim、Kobayashi Shimpei、Ye Zi
  • Journal Title: Geometriae Dedicata Volume: 216 Issue: 6 Pages: 71-71
  • DOI: 10.1007/s10711-022-00733-3
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Minimal surfaces with non-trivial geometry in the three-dimensional Heisenberg group (2022)
  • Author(s): Dorfmeister Josef F.、Inoguchi Jun-ichi、Kobayashi Shimpei
  • Journal Title: Complex Manifolds Volume: 9 Issue: 1 Pages: 285-336
  • DOI: 10.1515/coma-2021-0141
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] A characterization of the alpha-connections on the statistical manifold of normal distributions (2021)
  • Author(s): H. Furuhata, J. Inoguchi, S. Kobayashi
  • Journal Title: Information Geometry Volume: 4 Issue: 1 Pages: 177-188
  • DOI: 10.1007/s41884-020-00037-z
  • Peer Reviewed
• [Journal Article] Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space. (2021)
  • Author(s): Dorfmeister, Josef F., Kobayashi Shimpei
  • Journal Title: Annali di Matematica Pura ed Applicata. Series IV Volume: 200 Issue: 2 Pages: 521-546
  • DOI: 10.1007/s10231-020-01005-1
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Ruh-Vilms theorems for minimal surfaces without complex points and minimal Lagrangian surfaces in $\mathbb {C}P^2$ (2020)
  • Author(s): Dorfmeister Josef F., Kobayashi Shimpei, Ma Hui
  • Journal Title: Mathematische Zeitschrift Volume: 296 Issue: 3-4 Pages: 1751-1775
  • DOI: 10.1007/s00209-020-02497-6
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The Gauss maps of Demoulin surfaces with conformal coordinates (2020)
  • Author(s): Inoguchi Jun-ichi, Kobayashi Shimpei
  • Journal Title: Science China Mathematics Volume: - Issue: 7 Pages: 1479-8211
  • DOI: 10.1007/s11425-020-1738-0
  • Peer Reviewed
• [Journal Article] Representation formula for discrete indefinite affine spheres (2020)
  • Author(s): Kobayashi Shimpei、Matsuura Nozomu
  • Journal Title: Differential Geometry and its Applications Volume: 69 Pages: 101592-101592
  • DOI: 10.1016/j.difgeo.2020.101592
  • Peer Reviewed
• [Journal Article] Survey on real forms of the complex A2(2)-Toda equation and surface theory (2019)
  • Author(s): Dorfmeister Josef F.、Freyn Walter、Kobayashi Shimpei、Wang Erxiao
  • Journal Title: Complex Manifolds Volume: 6 Issue: 1 Pages: 194-227
  • DOI: 10.1515/coma-2019-0011
  • Peer Reviewed
• [Journal Article] General-affine invariants of plane curves and space curves (2019)
  • Author(s): Kobayashi Shimpei、Sasaki Takeshi
  • Journal Title: Czechoslovak Mathematical Journal Volume: 70 Issue: 1 Pages: 67-104
  • DOI: 10.21136/cmj.2019.0165-18
  • Peer Reviewed
• [Presentation] The generalized Weierstrass type representation for surfaces (1), (2) (2022)
  • Author(s): Shimpei Kobayashi
  • Organizer: The 13th MSJ-SI 2020 Mathematical Society of Japan - Seasonal Institute Differential Geometry and Integrable Systems - Mathematics of Symmetry, Stability and Moduli -
  • Int'l Joint Research / Invited
• [Presentation] 等質的な統計多様体について (2020)
  • Author(s): 小林真平，大野優
  • Organizer: ミニワークショップ 統計多様体の幾何学とその周辺 (12)
  • Invited
• [Presentation] 不定値複素射影平面の時間的全実極小曲面について (2019)
  • Author(s): 小林真平
  • Organizer: 北九州幾何学研究集会2019
  • Invited
• [Presentation] 一般アフィン平面および空間曲線について (2019)
  • Author(s): 小林真平
  • Organizer: 幾何学コロキウム（北海道大学）
  • Invited
• [Presentation] A real form of a_2^(2) and timelike minimal Lagrangian surfaces (2018)
  • Author(s): Shimpei Kobayashi
  • Organizer: Geometry Seminar at University of Augsburg
  • Invited
• [Presentation] A real form of a_2^(2) and timelike minimal Lagrangian surfaces (2018)
  • Author(s): Shimpei Kobayashi
  • Organizer: Integrable geometry at Bayrischzell
• [Presentation] D'Alembert type representation for discrete surfaces (2018)
  • Author(s): Shimpei Kobayashi
  • Organizer: SFB Colloquium; Technical University of Berlin
  • Invited
• [Presentation] Minimal surfaces in the 3-dimensional Heisenberg group (2018)
  • Author(s): Shimpei Kobayashi
  • Organizer: m:iv Summer 2018 Workshop
  • Int'l Joint Research / Invited
• [Presentation] General-affine invariants of plane curves and space curves (2018)
  • Author(s): Shimpei Kobayashi
  • Organizer: Differential Geometry in honor of Udo Simon's 80th birthday
  • Int'l Joint Research
• [Presentation] 離散微分幾何学と可積分系 (2018)
  • Author(s): 小林 真平
  • Organizer: 日本数学会 北海道支部会
  • Invited
• [Remarks] Kobayashi Shimpei's Homepage
  • URL: https://sites.google.com/site/kobayashishimpeisite/
• [Remarks] Kobayashi Shimpei's Homepage
  • URL: https://sites.google.com/site/kobayashishimpeisite