2023 Fiscal Year Final Research Report
Research on integrable geometry and submanifolds
| Project/Area Number |
21K03214
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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 | Tohoku University |
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Principal Investigator |
Miyaoka Reiko 東北大学, 理学研究科, 名誉教授 (70108182)
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| Project Period (FY) |
2021-04-01 – 2024-03-31
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| Keywords | 等径超曲面 / ガウス写像 / フレア理論 / ラグランジュ部分多様体 / 極小曲面 / ２次元戸田方程式 / 除外値問題 / Chern予想 |
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| Outline of Final Research Achievements |
Regarding the unresolved part of the Floer homology of the Gauss image L of a spherical isoparametric hypersurface N, we made the calculation of the topology of N applicable to more general spaces, and partially solved the calculation of the cohomolology of L. Particularly in the case related to the Clifford algebra containing an infinite number of non-homogeneous examples, we have even calculated the cohomology of L in some cases.This is a collaborative research with Yoshihiro Ohnita of Osaka Metropolitan University and Hiroshi Irie of Ibaraki University.
Regarding minimal surfaces, we have conducted research on metrically characterizing minimal Lagrangian surfaces in complex projective spaces, using periodic solutions of some integrable equation. We have also worked on the excluded value problem of Gauss maps of algebraic minimal surfaces.
Recently, we are studying the Chern conjecture in the case of Dupin hypersurfaces.
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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 |
Arnold予想の解決につながるフレア理論において，フレアホモロジーは特殊な場合にしか計算されておらず，非等質例を無限に含む等径超曲面のガウス像は好い対象であるが，その位相は複雑なので，その未解決部分に挑戦をしている．
極小曲面論は可積分系理論との関連において，曲面の計量のみたす２次元戸田方程式の解との関係が重要である．以前得た周期解と曲面の構造の関係を，複素射影空間の極小ラグランジュ曲面に適用すると興味深い結果が得られる．また代数的極小曲面のガウス写像の除外値問題は難解でライフワークとして取り組んでいる．
Chern予想にDupin超曲面で取り組むことは斬新である．
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