Research of submanifolds in symmetric spaces by usingthe infinite dimensional geometry and the complexification

• Project/Area Number: 21540095
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• 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, 准教授 (00281410)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 部分多様体幾何 / リ-群作用 / 平均曲率流 / 対称空間 / 無限次元幾何 / 複素化 / 幾何学

Research Abstract

Main results of this research are as follows.
(1) We showed that complex equifocal submanifolds in a symmetric space of non-compact type are congruent to principal orbits of Lie group actions called “Hermann action” under certain conditions.
(2) We showed that non-minimal equifocal submanifolds in a symmetric space of compact type collapse to their focal submanifolds along the mean curvature flow.
(3) We investigated the regularized mean curvature flow for invariant hypersurfaces in a Hilbert space equipped with a Hilbert Lie group free action and proved a certain kind of strongly convex preservability theorem for the flow.
(4) We classified hyperpolar actions on symmetric spaces of non-compact type under some conditions.

Research Products

• [Journal Article] Collapse of the mean curva- ture flow for equifocal submanifolds (2011)
  • Author(s): 小池直之
  • Journal Title: Asian Journal of Mathematics Volume: Vol.15, No.1 Pages: 101-128
  • Peer Reviewed
• [Journal Article] Parabolic subgroups of semi- simple Lie groups and Einstein solv- manifolds (2011)
  • Author(s): 田丸博士
  • Journal Title: Mathematische Annalen Volume: Vol.351, No.1 Pages: 51-66
  • Peer Reviewed
• [Journal Article] Collapse of the mean curvature flow for equifocal submanifolds (2011)
  • Author(s): 小池直之
  • Journal Title: Asian Journal of Mathematics Volume: 15 Pages: 101-128
  • Peer Reviewed
• [Journal Article] Hermann type actions on, Vol.34,No.2, 2010, pp. (5)a pseudo-Riemannian symmetric space (2010)
  • Author(s): 小池直之
  • Journal Title: Tsukuba Journal of Mathema- tics Volume: Vol.34,No.2
  • Peer Reviewed
• [Journal Article] On cuvature-adapted and Proper complex equifocal submani- folds (2010)
  • Author(s): 小池直之
  • Journal Title: Kyungpook Mathenmatical Journal Volume: Vol.50, No.4 Pages: 509-536
  • Peer Reviewed
• [Journal Article] Examples of a complex hperpolar action without singular orbit (2010)
  • Author(s): 小池直之
  • Journal Title: Cubo A Mathematical Journal Volume: Vol.12, No.2 Pages: 131-147
  • Peer Reviewed
• [Journal Article] The homogeneous slice theo- rem for the complete complexification of a proper complex equifocal subma- nifold (2010)
  • Author(s): 小池直之
  • Journal Title: Tokyo Journal of Mathematics Volume: Vol.33, No.1 Pages: 1-30
  • Peer Reviewed
• [Journal Article] Hyperpolar homogene- ous foliations on symmetric spaces of noncompact type (2010)
  • Author(s): 田丸博士 ,Jurgen Berndt, Jose Carlos Diaz-Ramos
  • Journal Title: Journal of Differen- tial Geometry Volume: Vol.86, No.2 Pages: 191-235
  • Peer Reviewed
• [Journal Article] The homogeneous slice theorem for the complete complexification of a proper complex equifocal submanifold (2010)
  • Author(s): 小池直之
  • Journal Title: Tokyo Journal of Mathematics Volume: 33 Pages: 1-30
  • Peer Reviewed
• [Journal Article] Example of a complex hyperpolar action without singular orbit (2010)
  • Author(s): 小池直之
  • Journal Title: Cubo A Mathematical Journal Volume: 12 Pages: 131-147
  • Peer Reviewed
• [Journal Article] On curvature-adapted and proper complex equifocal submanifolds (2010)
  • Author(s): 小池直之
  • Journal Title: Kyungpook Mathematical Journal Volume: 50 Pages: 509-536
  • Peer Reviewed
• [Journal Article] Hermann type actions on a pseudo-Riemannian symmetric space (2010)
  • Author(s): 小池直之
  • Journal Title: Tsukuba Journal of Mathematics Volume: 34 Pages: 137-172
  • NAID: 120006309485
  • Peer Reviewed
• [Journal Article] The homogeneous slice theorem for the complete complexification of a proper complex equifocal submanifold (2010)
  • Author(s): 小池直之
  • Journal Title: Tokyo Journal of Mathematics 33 Pages: 1-30
  • Peer Reviewed
• [Journal Article] Cohomo- geneity one actions on symmetric sp- aces of noncompact type
  • Author(s): 田丸博士,Jurgen Berndt
  • Journal Title: Journal fur Reine und Angewandte Mathematik
  • Peer Reviewed
• [Presentation] Certain kind of isopara- metric submanifolds in symmetric spa- ces of non-compact type and Hermann actions (2013)
  • Author(s): 小池直之
  • Organizer: The 6th OCAMI-KNUGRG Joint Differentail Geometry Workshop on Sub- manifold Theory in Symmetric spaces and Lie theory in Finite and Infinite Dimensions
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2013-02-02
• [Presentation] On the moduli space of left-invariant metrics on a Lie group (2013)
  • Author(s): 田丸博士
  • Organizer: The 6^<th> OCAMI-KNUGRG Joint Differential Geometry Workshop on Submanifold Theory in Symmetric spaces and Lie theory in Finite and Infinite Dimensions (招待講演)
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2013-02-02
• [Presentation] Certain kind of isoparametric submanifolds in symmetric spaces of non-compact type and Hermann actions (2013)
  • Author(s): 小池直之
  • Organizer: The 6th OCAMI-KNUGRG Joint Differential Geometry Workshop on Submanifold Theory in Symmetric spaces and Lie Theory in Finite and Infinite Dimensions
  • Place of Presentation: 大阪市立大学
  • Invited
• [Presentation] 非コンパクト型対称空間内のある種の等径超曲面の分類 (2013)
  • Author(s): 小池直之
  • Organizer: 日本数学会2013年度会
  • Place of Presentation: 京都大学
• [Presentation] 対称空間内の等径部分多様体を発する平均曲率流 (2012)
  • Author(s): 小池直之
  • Organizer: 広島幾何学研究集会 2012
  • Place of Presentation: 広島大学
  • Year and Date: 2012-10-04
• [Presentation] 非コンパクト型対称空間内のある種の等径部分多様体の分類 (2012)
  • Author(s): 小池直之
  • Organizer: 部分多様体の幾何とリー群作用 2012 (招待講演)
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-09-04
• [Presentation] 非コンパクト対称空間への群作用とその応用 (2012)
  • Author(s): 田丸博士
  • Organizer: 第59回幾何学シンポジウム
  • Place of Presentation: 九州大学
  • Year and Date: 2012-08-27
• [Presentation] 非コンパクト型対称空間内のある種の等径部分多様体の分類 (2012)
  • Author(s): 小池直之
  • Organizer: 部分多様体の微分幾何学及び関連課題-前田定廣先生還暦記念研究集会
  • Place of Presentation: 佐賀大学
  • Year and Date: 2012-08-04
• [Presentation] 複素双曲空間内の等質部分多様体に関するいくつかの話題 (2012)
  • Author(s): 田丸博士
  • Organizer: 部分多様体の微分幾何学及び関連課題-前田定廣先生還暦記念研究集会(招待講演)
  • Place of Presentation: 佐賀大学
  • Year and Date: 2012-08-04
• [Presentation] 対称空間における非等方的エネルギー汎関数および非等方的等径超曲面 (2012)
  • Author(s): 小池直之
  • Organizer: 日本数学会 2012年度会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-26
• [Presentation] 対称空間における非等方的エネルギー汎関数および非等方的等径超曲面 (2012)
  • Author(s): 小池直之
  • Organizer: 日本数学会2012年度会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-26
• [Presentation] 非コンパクト型対称空間内のある種の等径部分多様体の分類 (2012)
  • Author(s): 小池直之
  • Organizer: 部分多様体の微分幾何学及び関連課題 (前田定廣先生還暦記念研究集会)
  • Place of Presentation: 佐賀大学
  • Invited
• [Presentation] 非コンパクト型対称空間内のある種の等径部分多様体の分類 (2012)
  • Author(s): 小池直之
  • Organizer: 部分多様体幾何とリー群作用2012
  • Place of Presentation: 東京理科大学
• [Presentation] 対称空間内の等径部分多様体を発する平均曲率流 (2012)
  • Author(s): 小池直之
  • Organizer: 広島幾何学研究集会２０１２
  • Place of Presentation: 広島大学
• [Presentation] ヒルベルト空間内のある種の不変超曲面を発する平均曲率流について,日本数学会 (2011)
  • Author(s): 小池直之
  • Organizer: 2011 年度秋季総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-30
• [Presentation] ヒルベルト空間内のある種の不変超曲面を発する平均曲率流について (2011)
  • Author(s): 小池直之
  • Organizer: 日本数学会2011年度秋季総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-30
• [Presentation] Parabolic subgroups of semi- simple Lie groups and submanifold geo- metry, DFG-JSPS Seminar (2011)
  • Author(s): 田丸博士
  • Organizer: Lie groups: Geometry and Analysis
  • Place of Presentation: Paderborn
  • Year and Date: 2011-09-09
• [Presentation] Parabolic subgroups and sub- manifold geometry of noncompact symme- tric spaces (2011)
  • Author(s): 田丸博士
  • Organizer: International Mini-Work shop on Geometry of Symmetric Spaces and Related Fields
  • Place of Presentation: Kyungpook National University (韓国)
  • Year and Date: 2011-04-14
• [Presentation] Hermann型作用の主軌道の部分多様体幾何的特徴づけ (2011)
  • Author(s): 小池直之
  • Organizer: 日本数学会2011年度会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-21
• [Presentation] プロパー複素等焦部分多様体と平坦な切断をもつ等径部分多様体 (2011)
  • Author(s): 小池直之
  • Organizer: 日本数学会2011年度会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-21
• [Presentation] Parabolic subgroups and geo- metry of solvable Lie groups (2011)
  • Author(s): 田丸博士
  • Organizer: 10th Pan- hellenic Geometry Conference
  • Place of Presentation: Patras (ギリシャ)
• [Presentation] On the indices of minimal orbits of Hermann actions (2010)
  • Author(s): 小池直之
  • Organizer: 日本数学会2010年度秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] 非コンパクト対称空間への良い作用の構成法 (2010)
  • Author(s): 田丸博士
  • Organizer: 部分多様体幾何とリー群作用(招待講演)
  • Place of Presentation: 東京理科大学
  • Year and Date: 2010-09-08
• [Presentation] O n equifocal submanifolds and complex equifocal submanifolds (2010)
  • Author(s): 小池直之
  • Organizer: Workshop on Hypersurfaces Geometry and Integrable Systems
  • Place of Presentation: 東北大学
  • Year and Date: 2010-08-27
• [Presentation] On equifocal submanifolds and complex equifocal submanifolds (2010)
  • Author(s): 小池直之
  • Organizer: Workshop on Hypersurfaces Geometry and Integrable systems
  • Place of Presentation: 東北大学(招待講演)
  • Year and Date: 2010-08-27
• [Presentation] Cohomogeneity one actions on symmetric spaces of noncompact type (II) (2010)
  • Author(s): 田丸博士
  • Organizer: Workshop on Hypersurfaces Geome- try and Integrable Systems
  • Place of Presentation: 東北大学
  • Year and Date: 2010-08-26
• [Presentation] プロパー複素等焦部分多様体を初期データにもつ平均曲率流 (2010)
  • Author(s): 小池直之
  • Organizer: 日本数学会 2010年度会
  • Place of Presentation: 慶応大学
  • Year and Date: 2010-03-24
• [Presentation] プロパー複素等焦部分多様体を初期データにもつ平均曲率流 (2010)
  • Author(s): 小池直之
  • Organizer: 日本数学会2010年度会
  • Place of Presentation: 慶応大学
  • Year and Date: 2010-03-24
• [Presentation] Collapse of the mean curva- ture flows of equifocal submanifolds and proper comlex equifocal submani- folds(招待講演) (2010)
  • Author(s): 小池直之
  • Organizer: The 2nd TIMS-OCAMI Joint International Workshop on Diffe- rential Geometry and Geometric Ana- lysis
  • Place of Presentation: National Tiwan University(台湾)
  • Year and Date: 2010-03-21
• [Presentation] Collapses of the mean curvature flows for equifocal submanifolds and proper complex equifocal submanifolds (2010)
  • Author(s): 小池直之
  • Organizer: 第2回台大数学科学中心-大阪市立大学数学研究所共催国際ワークショップ「微分幾何学と幾何解析」
  • Place of Presentation: 国立台湾大学
  • Year and Date: 2010-03-21
• [Presentation] Homogeneity of an equifocal submanifold and a proper complex equi- focal submanifold I,II (2010)
  • Author(s): 小池直之
  • Organizer: The 14th Inter- national Workshop on Differential Geo- metry and The 3rd KNUGRG-OCAMI Diffe- rential Geometry Workshop
  • Place of Presentation: Kyungpook National University(韓国)
• [Presentation] Homogeneous submanifolds in noncompact symmetric spaces(全 6回) (2010)
  • Author(s): 田丸博士
  • Organizer: International Mini Workhop on Integral Geometry and Symmetric Spaces
  • Place of Presentation: Kyungpook National University (韓国)
• [Presentation] Homogeneity of an equifocal submanifold and a proper complex equifocal submanifold I, II (2010)
  • Author(s): 小池直之
  • Organizer: The 14^<th> International Workshop on Differential Geometry and The 3^<rd> KNUGRG-OCAMI Differential Geometry Workshop
  • Place of Presentation: Kyungpook National University(招待講演)
• [Presentation] The mean curvature flow for equifocal submanifolds (2009)
  • Author(s): 小池直之
  • Organizer: The 1st KNUGRG -OCAMI Differential Geometry Workshop
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2009-10-29
• [Presentation] Collapse of the mean curvature flow for equifocal submanifolds (2009)
  • Author(s): 小池直之
  • Organizer: 大阪市立大学-慶北大学共催研究集会「部分多様体幾何学とリー群論」
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2009-10-29
• [Presentation] 等焦部分多様体を初期データ にもつ平均曲率流 (2009)
  • Author(s): 小池直之
  • Organizer: 2010 年度秋季総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] 等焦部分多様体を初期データにもつ平均曲率流 (2009)
  • Author(s): 小池直之
  • Organizer: 日本数学会2009年秋季総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-24
• [Presentation] 等焦部分多様体を初期データ にもつ平均曲率流 (2009)
  • Author(s): 小池直之
  • Organizer: RIMS 研究集会「部分多様体論とその周辺領域における新たな研究対象」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-23
• [Presentation] 等焦部分多様体を初期データにもつ平均曲率流 (2009)
  • Author(s): 小池直之
  • Organizer: 京都大学数理解析研究所RIMS研究集会「部分多様体論とその周辺領域における新たな研究対象」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-23