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

• 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
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• 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 / 等焦部分多様体 / カラビ・ヤウ多様体 / 部分多様体 / 逆平均曲率流 / Polar作用 / Kac-Moody型無限次元対称空間

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.

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。

Research Products

• [Journal Article] CLASSIFICATION OF ISOPARAMETRIC SUBMANIFOLDS ADMITTING A REFLECTIVE FOCAL SUBMANIFOLD IN SYMMETRIC SPACES OF NON-COMPACT TYPE (2020)
  • Author(s): Koike Naoyuki
  • Journal Title: Osaka Journal of Mathematics Volume: 57 Issue: 1 Pages: 207-246
  • DOI: 10.18910/73746
  • NAID: 120006786533
  • ISSN: 00306126
  • URL: https://ocu-omu.repo.nii.ac.jp/records/2010771
  • Peer Reviewed / Open Access
• [Journal Article] Calabi-Yau structures and special Lagrangian submanifolds of complexified symmetric spaces (2019)
  • Author(s): Koike Naoyuki
  • Journal Title: Illinois Journal of Mathematics Volume: 63 Issue: 4 Pages: 575-600
  • DOI: 10.1215/00192082-8018607
  • Peer Reviewed
• [Journal Article] Gauss maps of the Ricci-mean curvature flow (2018)
  • Author(s): Naoyuki Koike, Hikaru Yamamoto
  • Journal Title: Geometriae Dedicata Volume: 194 Issue: 1 Pages: 169-185
  • DOI: 10.1007/s10711-017-0271-8
  • Peer Reviewed
• [Journal Article] A modified mean curvature flow in Euclidean space and soap bubbles in symmetric spaces (2018)
  • Author(s): Naoyuki Koike
  • Journal Title: Geometriae Dedicata Volume: 195 Issue: 1 Pages: 1-17
  • DOI: 10.1007/s10711-017-0273-6
  • Peer Reviewed
• [Journal Article] Mean curvature flow of certain kind of isoparametric foliation on non-compact symmetric spaces (2018)
  • Author(s): Naoyuki Koike
  • Journal Title: CUBO A Mathematical Journal Volume: 20 Issue: 3 Pages: 13-30
  • DOI: 10.4067/s0719-06462018000300013
  • Peer Reviewed
• [Presentation] The existence and the uniqueness of regularized mean curvature flows (2021)
  • Author(s): Naoyuki Koike
  • Organizer: International Workshop on Geometric Evolution Equations and Related Fields
  • Int'l Joint Research / Invited
• [Presentation] Calabi-Yau structures and Special Lagrangian submanifolds of complexified symmetric spaces (2020)
  • Author(s): Naoyuki Koike
  • Organizer: The 18th OCAMI-RIRCM Joint Differential Geometry Workshop ``Differential Geometry of Submanifolds in Symmetric Spaces and Related Problems’’(日本学術振興会 二国間交流事業 韓国 (NRF) )
  • Int'l Joint Research / Invited
• [Presentation] Isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type (2019)
  • Author(s): Naoyuki Koike
  • Organizer: Workshop on the isoparametric theory
  • Int'l Joint Research / Invited
• [Presentation] Regularized mean curvature flow in a Hilbert space and its application to the Gauge theory (2019)
  • Author(s): Naoyuki Koike
  • Organizer: The 22 Workshop on Differential Geometry of Submanifolds in Symmetric Spaces and The 17th RIRCM-OCAMI Joint Differential Geometry
  • Int'l Joint Research / Invited
• [Presentation] Regularized mean curvature flow in a Hilbert space and its application to the Gauge theory (2019)
  • Author(s): Naoyuki Koike
  • Organizer: Symmetry and Shape (Celebrating the 60th birthday of Prof. J. Berndt)
  • Int'l Joint Research
• [Presentation] Regularized mean curvature flow in a Hilbert space and its application to the gauge theory (2019)
  • Author(s): Naoyuki Koike
  • Organizer: RIMS研究集会「非線形偏微分方程式における定性的理論」
  • Int'l Joint Research / Invited
• [Presentation] Regularized mean curvature flow in a Hilbert space and its application to the gauge theory (2019)
  • Author(s): 小池直之
  • Organizer: 日本数学会
• [Book] 理論物理に潜む部分多様体幾何 (2021)
  • Author(s): 小池 直之
  • Total Pages: 440
  • Publisher: 共立出版
  • ISBN: 9784320114401
• [Book] 平均曲率流-部分多様体の時間発展- (2019)
  • Author(s): 小池直之
  • Total Pages: 376
  • Publisher: 共立出版
  • ISBN: 9784320113763
• [Remarks] 小池直之研究室
  • URL: https://www.rs.kagu.tus.ac.jp/~koike/