Research of submanifolds in symmetric spaces and their time evolution along various curvature flows

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03311
• 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: Tokyo University of Science
• Principal Investigator: Koike Naoyuki 東京理科大学, 理学部第一部数学科, 教授 (00281410)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 平均曲率流 / 対称空間 / 等径部分多様体 / 複素等焦部分多様体 / 固有フレッドホルム部分多様体 / ゲージ理論 / カラビ・ヤウ構造 / 特殊ラグランジュ部分多様体

Outline of Final Research Achievements

Main research results in this research subject are as follows. First, we proved the homogeneity theorem for isoparametric submanifolds in a symmetric spaces of non-compact type admitting a reflective focal submanifold by using the complexification and the linearization to an infinite dimensional Hilbert space.
Secondly we proved a certain kind of collapsing theorem for the invariant regularized mean curvature flow in a Hilbert space equipped with a certain kind of Hilbert Lie group action and made the theory based to apply the collapsing theorem to the Gauge theory. Thirdly we gave a construction of Calabi-Yau structures on the complexification of a symmetric space of compact type and a construction of special Lagrangian submanifolds in the Calabi-Yau manifold.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究課題における研究成果は，微分幾何学の見地から，ゲージ理論や超対称性理論をはじめとする理論物理学を研究する上で，重要な結果になるのではないかと考えている。特に，研究成果の一つである正則化された平均曲率流の研究をゲージ理論へ応用するために土台となる理論の構築は，今後，注目されるのではないかと考えている。