Research of submanifolds and mean curvature flows in symmetric spaces by using the infinite dimensional geometry and the complexification

• Project/Area Number: 25400076
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• 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, 教授 (00281410)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 部分多様体幾何 / 無限次元幾何 / 平均曲率流 / リー群作用 / 対称空間 / 部分多様体の複素化 / 等焦部分多様体 / 無限次元等径部分多様体 / 対称空間の複素化 / Polar作用 / 体積を保存する平均曲率流 / 正則化された平均曲率流 / ヒルベルト空間 / 等径部分多様体 / 無限次元部分多様体幾何 / 幾何学

Outline of Final Research Achievements

Main results of this reseach are as follows. First we obtained the homogeneity theorem for ant-Kaehler isoparametric submanfolds in the infnite dimensional anti-Kaehler space under the assumption of a certain kind of diagonalzability of the shape operators and furthermore, completed almost the proof of the homogeneity theorem for certain kind of complex equifocal submanifolds in symmetric spaces of non-compact type by using the result. Secondly we obatined the collapsing theorem for the regularized mean curvature flow starting from horizontally convex invariant hypersurfaces in a Hilbert space. Thirdly we obtained a result about the volume-preserving mean curvature flow starting from tubes of nonconstant radius over certain reflective submanifold in rank one symmetric spaces of non-compact type.

Research Products

• [Journal Article] Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type (2017)
  • Author(s): Naoyuki Koike
  • Journal Title: Calculus of Variations and Partial Differential Equations Volume: 56 Issue: 3 Pages: 1-51
  • DOI: 10.1007/s00526-017-1156-2
  • Peer Reviewed
• [Journal Article] The mean curvature flow for invariant hypersurfaces in a Hilbert space with an almost free group action (2017)
  • Author(s): Naoyuki Koike
  • Journal Title: Asian Journal of Mathematics Volume: 21
  • Peer Reviewed
• [Journal Article] A holonomy invariant anisotropic surface nergy in a Riemannian manifold (2016)
  • Author(s): Naoyuki Koike
  • Journal Title: Differential Geometry and its Applications Volume: 44 Pages: 98-121
  • DOI: 10.1016/j.difgeo.2015.11.003
  • Peer Reviewed
• [Journal Article] On the indices of minimal orbits of Hermann actions (2015)
  • Author(s): Naoyuki Koike
  • Journal Title: Hokkaido Mathematical Journal Volume: 44
  • Peer Reviewed
• [Journal Article] The inverse mean curvature flow in rank one symmetric spaces of non-compact type (2015)
  • Author(s): Naoyuki Koike and Yusuke Sakai
  • Journal Title: Kyushu Journal of Mathematics Volume: 69 Issue: 2 Pages: 259-284
  • DOI: 10.2206/kyushujm.69.259
  • NAID: 130005103466
  • Peer Reviewed
• [Journal Article] The constancy of the principal curvatures of curvature-adapted submanifolds in symmetric spaces (2014)
  • Author(s): Naoyuki Koike
  • Journal Title: Differential Geometry and its Applications Volume: 35 Pages: 103-113
  • DOI: 10.1016/j.difgeo.2014.06.004
  • Peer Reviewed
• [Journal Article] Research of submanifolds in symmetric spaces by using the complexification and the infinite dimensional geometry (2014)
  • Author(s): Naoyuki Koike
  • Journal Title: SUT Journal of Mathematics Volume: 50 Pages: 103-129
  • Peer Reviewed
• [Journal Article] The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry (2014)
  • Author(s): Naoyuki Koike
  • Journal Title: SUT Journal of Mathematics Volume: 50 Pages: 271-295
  • Peer Reviewed
• [Journal Article] Examples of certain kind of minimal orbits of Hermann actions (2014)
  • Author(s): Naoyuki Koike
  • Journal Title: Hokkaido Mathematical Journal Volume: 43 Pages: 21-42
  • Peer Reviewed
• [Journal Article] A Cartan type identity for isoparametric hypersurfaces in symmetric spaces (2014)
  • Author(s): Naoyuki Koike
  • Journal Title: Tohoku Mathematical Journal Volume: 66
  • NAID: 130007953687
  • Peer Reviewed
• [Journal Article] Collapse of the mean curvature flow for isoparametric submanifolds in non-compact symmetric spaces (2014)
  • Author(s): Naoyuki Koike
  • Journal Title: Kodai Mathematical Journal Volume: 37
  • NAID: 130004948997
  • Peer Reviewed
• [Journal Article] On the indices of minimal orbits of Hermann actions (2014)
  • Author(s): Naoyuki Koike
  • Journal Title: Hokkaido Mathematical Journal Volume: 43
  • Peer Reviewed
• [Journal Article] Homogeneity of infinite dimensional anti-Kaehler isoparametric submanifolds (2014)
  • Author(s): Naoyuki Koike
  • Journal Title: Tokyo Journal of Mathematics Volume: 37
  • Peer Reviewed
• [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の崩壊について (2017)
  • Author(s): Naoyuki Koike
  • Organizer: 日本数学会2017年度会
  • Place of Presentation: 首都大学東京
• [Presentation] Collapse of the regularized mean curvature flow for invariant hypersurfaces in a Hilbert space (2017)
  • Author(s): Naoyuki Koike
  • Organizer: The 13th OCAMI-RIRCM Joint Differential Geometry Workshop on Submanifold Geometry and Lie Theory
  • Place of Presentation: 早稲田大学
  • Invited
• [Presentation] Volume-preserving mean curvature flow for tubes in symmetric spaces (2016)
  • Author(s): Naoyuki Koike
  • Organizer: The 11th OCAMI-RIRCM Joint Differential geometry Workshop on Submanifolds and Lie theory
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2016-03-20
  • Invited
• [Presentation] 平均曲率流に沿うcurvature-adapted性の保存性について (2016)
  • Author(s): Naoyuki Koike
  • Organizer: 日本数学会2016年度会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-18
• [Presentation] 無限次元線形空間へのリフトを利用した平均曲率流の研究について (2016)
  • Author(s): Naoyuki Koike
  • Organizer: Geometric flows and related problems
  • Place of Presentation: 東京工業大学
  • Year and Date: 2016-03-04
  • Invited
• [Presentation] Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type (2016)
  • Author(s): Naoyuki Koike
  • Organizer: RIMS研究集会「Regularity, singularity and long time behavior for partial differential equations with conservation law
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Volume-preserving mean curvature flow for tubes in rank one symmetric space of non-compact type (2016)
  • Author(s): Naoyuki Koike
  • Organizer: The 20th International Workshop on Hermitian Symmetric Spaces and Submanifolds and 12th OCAMI-RIRCM Joint Differential Geometry Workshop on Submanifolds and Lie Theory
  • Place of Presentation: Kyungpook National University, Korea
  • Invited
• [Presentation] 階数1非コンパクト対称空間内のチューブを発する体積を保存する平均曲率流の長時間挙動 (2016)
  • Author(s): Naoyuki Koike
  • Organizer: 日本数学会2016年度秋季総合分科会
  • Place of Presentation: 関西大学
• [Presentation] Volume-preserving mean curvature flow for tubes with boundary in symmetric spaces (2016)
  • Author(s): Naoyuki Koike
  • Organizer: The First Japan-Taiwan Joint Conference on Differential Geometry & The 8th TIMS-OCAMI-Waseda Joint International Workshop on Differential Geometry and Geometric Analysis
  • Place of Presentation: 早稲田大学
  • Invited
• [Presentation] リーマン多様体における平均曲率流の無限次元線形空間へのリフトについて (2015)
  • Author(s): Naoyuki Koike
  • Organizer: 第138回神楽坂幾何セミナー
  • Place of Presentation: 東京理科大学
  • Year and Date: 2015-11-28
• [Presentation] 対称空間内のチューブを発する体積を保存する平均曲率流 (2015)
  • Author(s): Naoyuki Koike
  • Organizer: 日本数学会2015年度秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] 無限次元アンチケーラー等径部分多様体のバナッハリー群作用による等質性 (2015)
  • Author(s): Naoyuki Koike
  • Organizer: 日本数学会2015年度会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-22
• [Presentation] ヒルベルト空間内の不変超曲面を発する正則化された平均曲率流 (2014)
  • Author(s): Naoyuki Koike
  • Organizer: 部分多様体幾何とリー群作用2014
  • Place of Presentation: 東京理科大学森戸記念館
  • Year and Date: 2014-09-05
• [Presentation] The Regularized Mean Curvature Flow for Invariant Hypersurfaces in a Hilbert Space (2014)
  • Author(s): Naoyuki Koike
  • Organizer: Seoul ICM 2014 (International Congress of Mathematicians)-Satellite Conference on Real and Complex Submanifolds
  • Place of Presentation: National Institute for Mathematical Science, Korea
  • Year and Date: 2014-08-10 – 2014-08-12
  • Invited
• [Presentation] The Regularized Mean Curvature Folw for Invariant Hypersurfaces in a Hilbert Space with an Almost Free Lie Group Action (2014)
  • Author(s): Naoyuki Koike
  • Organizer: RIMS研究集会「群作用と部分多様体論の展開」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2014-06-26
  • Invited
• [Presentation] 概自由なリー群作用をもつヒルベルト空間内の不変な超曲面を発する平均曲率流 (2014)
  • Author(s): 小池直之
  • Organizer: 日本数学会2014年度会
  • Place of Presentation: 学習院大学
• [Presentation] Ｔｈｅ ｍｅａｎ ｃｕｒｖａｔｕｒｅ ｆｌｏｗ ｆｏｒ ａ ｃｏｎｖｅｘ ｈｙｐｅｒｓｕｒｆａｃｅ (2013)
  • Author(s): 小池直之
  • Organizer: Mini International Workshop on Quaternionic hyperbｏｌｉｃ ｓｐａｃｅ， Ｏｓｓｅｒｍａｎｎ ４－ｍａｎｉｆｏｌｄｓ ａｎｄ ｍｅａｎ ｃｕｒｖａｔure ｆｌｏｗ
  • Place of Presentation: Ｋｙｕｎｇｐｏｏｋ Ｎａｔｉｏｎａｌ Ｕｎｉｖｅｒｓｉｔｙ, Korea
  • Invited
• [Presentation] The mean curvature flow for a mean convex hypersurface (2013)
  • Author(s): 小池直之
  • Organizer: Mini International Workshop on Quaternionic hyperbｏｌｉｃ ｓｐａｃｅ， Ｏｓｓｅｒｍａｎｎ ４－ｍａｎｉｆｏｌｄｓ ａｎｄ ｍｅａｎ ｃｕｒｖａｔure ｆｌｏｗ
  • Place of Presentation: Ｋｙｕｎｇｐｏｏｋ Ｎａｔｉｏｎａｌ Ｕｎｉｖｅｒｓｉｔｙ, Korea
  • Invited
• [Presentation] The mean curvature flow for an equifocal submanifold (2013)
  • Author(s): 小池直之
  • Organizer: Mini International Workshop on Quaternionic hyperbｏｌｉｃ ｓｐａｃｅ， Ｏｓｓｅｒｍａｎｎ ４－ｍａｎｉｆｏｌｄｓ ａｎｄ ｍｅａｎ ｃｕｒｖａｔure ｆｌｏｗ
  • Place of Presentation: Ｋｙｕｎｇｐｏｏｋ Ｎａｔｉｏｎａｌ Ｕｎｉｖｅｒｓｉｔｙ, Korea
  • Invited
• [Presentation] The mean curvature flow for a complex equifocal submanifold (2013)
  • Author(s): 小池直之
  • Organizer: Mini International Workshop on Quaternionic hyperbｏｌｉｃ ｓｐａｃｅ， Ｏｓｓｅｒｍａｎｎ ４－ｍａｎｉｆｏｌｄｓ ａｎｄ ｍｅａｎ ｃｕｒｖａｔure ｆｌｏｗ
  • Place of Presentation: Ｋｙｕｎｇｐｏｏｋ Ｎａｔｉｏｎａｌ Ｕｎｉｖｅｒｓｉｔｙ, Korea
  • Invited
• [Presentation] The regularized mean curvature flow for an invariant hypersurface in a Hilbert space (2013)
  • Author(s): 小池直之
  • Organizer: Mini International Workshop on Quaternionic hyperbｏｌｉｃ ｓｐａｃｅ， Ｏｓｓｅｒｍａｎｎ ４－ｍａｎｉｆｏｌｄｓ ａｎｄ ｍｅａｎ ｃｕｒｖａｔure ｆｌｏｗ
  • Place of Presentation: Kyungpook National University, Korea
  • Invited