2022 Fiscal Year Final Research Report
Fusion of discrete and smooth integrable geometry
| Project/Area Number |
18K03265
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Hokkaido University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| 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 |
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| 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.
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| Free Research Field |
微分幾何学
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| 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．
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