New development of the integrable geometry

2021 Fiscal Year Final Research Report

• Project/Area Number: 15H03616
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Miyaoka Reiko 東北大学, 理学研究科, 名誉教授 (70108182)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 可積分幾何 / 等径超曲面 / ガウス写像 / フレア理論 / 交叉のハミルトン非解消性

Outline of Final Research Achievements

Starting from minimal surfaces, we developed the research into the theory of harmonic maps, which further goes to the theory of integrable systems through the 2D Toda equations. The theory of isoparametric hypersurfaces also appears as a geometry of wave fronts which concerns integrable systems. We solved one of the most difficult classification problems　in 2016, and the remaining case was solved by Q.S.Chi. The Gauss image L of isoparametric hypersurfaces gives a rich family of nice examples of minimal Lagrangian submanifolds in certain symplectic manifold, including infinitely many non-homogeneous ones. We are interested in their Floer homology, and proved the Hamiltonian non-displaceability of L except for 4 non-simply connected cases. We are now challenging these 4 remaining cases.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

フレア理論はシンプレクティック幾何のラグランジュ部分多様体のホモロジー論，そのハミルトン変形による交叉数を評価するArnold-Givental 予想で必要となる無限次元Morse理論として，Floerにより構築された．一般論としては深谷-Oh-太田-小野らが世界を牽引する研究を行っているが，フレアホモロジーの具体計算は特殊な場合を除き多くの困難を伴う．等径超曲面のガウス像という豊富な例は，非等質なものを無限に含むことから，非自明例として計算の価値がある．フレアホモロジーの計算が最終目標ではあるが，フレアホモロジーが定義できるか否かを定める重要な要件がハミルトン交叉非解消性である．