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New development of the theory of convergence and collapsing of Riemannian manifolds

Research Project

Project/Area Number 18H01118
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionUniversity of Tsukuba (2020-2021)
Kyoto University (2018-2019)

Principal Investigator

Yamaguchi Takao  筑波大学, 数理物質系(名誉教授), 名誉教授 (00182444)

Co-Investigator(Kenkyū-buntansha) 本多 正平  東北大学, 理学研究科, 教授 (60574738)
塩谷 隆  東北大学, 理学研究科, 教授 (90235507)
Project Period (FY) 2018-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥10,400,000 (Direct Cost: ¥8,000,000、Indirect Cost: ¥2,400,000)
Fiscal Year 2020: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2019: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Keywords崩壊理論 / 境界付きリーマン多様体 / アレクサンドロフ空間 / CAT(1)空間 / 境界特異点 / 崩壊 / 境界つき多様体 / スペクトル収束 / 境界付き多様体 / 負曲率リーマン多様体 / 測度距離空間 / 有界コホモロジー / 境界つきリーマン多様体 / 単体的体積 / CAT(1)-空間
Outline of Final Research Achievements

(1)We investigated the convergence and collapsing phenomena of Riemannian manifolds with boundary whose sectional curvatures are uniformly bounded below and the second fundamental forms of the boundaries are unifowmly bounded. We characterized the boundary singular points of the limit spaces, and determined the geometry of the limit spaces. (2)We determined the local structure of geodesically complete two-dimensional metric spaces with curvature bounded above, based on the topological singular point set. We also obtained an approximation theorem by polyhedral surfaces and the Gauss-Bonnet theorem.(3) We determined the topology of three-dimensional Alexandrov spaces with boundary.

Academic Significance and Societal Importance of the Research Achievements

閉じたリーマン多様体の崩壊理論は、Perelman によるポアンカレ予想・幾何化予想解決において重要な役割を果たした。一方で応用の観点からも境界付きリーマン多様体の崩壊の研究は重要であるが、Jeremy Wongの研究以後、我々の研究まで全く進展がなかった。我々の研究は新境地を開くものである。曲率が上に有界な空間は、幾何学的群論など関連分野は広い。しかしその局所構造は極めてワイルドなもので、これまで未解明であった。本研究により測地的に完備で曲率が上に有界な2次元距離空間の局所構造が世界で初めて解明された。これにより曲率の上界の根源的な本質を深く理解することが可能となった。

Report

(4 results)
  • 2021 Final Research Report ( PDF )
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • Research Products

    (32 results)

All 2021 2020 2019 2018 Other

All Int'l Joint Research (1 results) Journal Article (15 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 15 results,  Open Access: 2 results) Presentation (15 results) (of which Int'l Joint Research: 12 results,  Invited: 15 results) Book (1 results)

  • [Int'l Joint Research] Foshan University(中国)

    • Related Report
      2019 Annual Research Report
  • [Journal Article] New differential operator and non-collapsed RCD spaces2020

    • Author(s)
      Shouhei Honda
    • Journal Title

      Geom. Topol

      Volume: 24 Issue: 4 Pages: 2127-2148

    • DOI

      10.2140/gt.2020.24.2127

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Rigidity for positive Ricci curvature via metric measure geometry2020

    • Author(s)
      Shouhei Honda
    • Journal Title

      Adv. Stud. Pure Math.

      Volume: 85 Pages: 151-161

    • DOI

      10.2969/aspm/08510151

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Graph manifolds as ends of negatively curved Riemannian manifolds2020

    • Author(s)
      Fujiwara Koji、Shioya Takashi
    • Journal Title

      Geometry & Topology

      Volume: 24 Issue: 4 Pages: 2035-2074

    • DOI

      10.2140/gt.2020.24.2035

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Inradius collapsed manifolds.2019

    • Author(s)
      Yamaguchi, Takao; Zhang, Zhilang
    • Journal Title

      Geom. Topol.

      Volume: 23 Issue: 6 Pages: 2793-2860

    • DOI

      10.2140/gt.2019.23.2793

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Obtuse constants of Alexandrov spaces2019

    • Author(s)
      MITSUISHI Ayato、YAMAGUCHI Takao
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 71 Issue: 4 Pages: 1081-1103

    • DOI

      10.2969/jmsj/78917891

    • NAID

      130007733314

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Good coverings of Alexandrov spaces2019

    • Author(s)
      Mitsuishi Ayato、Yamaguchi Takao
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 372 Issue: 11 Pages: 8107-8130

    • DOI

      10.1090/tran/7849

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Isoperimetric rigidity and distributions of 1-Lipschitz functions2019

    • Author(s)
      Nakajima Hiroki、Shioya Takashi
    • Journal Title

      Advances in Mathematics

      Volume: 349 Pages: 1198-1233

    • DOI

      10.1016/j.aim.2019.04.043

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A note on collapsing geometry of hyperKahler four manifolds2019

    • Author(s)
      Shouhei Honda, Song Sun and Ruobing Zhang
    • Journal Title

      Sci. China Math.

      Volume: 62 Issue: 11 Pages: 2195-2210

    • DOI

      10.1007/s11425-019-1602-x

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Inradius collapsed manifolds2019

    • Author(s)
      T. Yamaguchi and Z. Zhang
    • Journal Title

      Geometry and Topology

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Lipschitz homotopy convergence of Alexandrov spaces2019

    • Author(s)
      A.Mitsuishi and T. Yamaguchi
    • Journal Title

      J. Geometric Analysis

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Good coverings of Alexandrov spaces2019

    • Author(s)
      A.Mitsuishi and T. Yamaguchi
    • Journal Title

      Trans. AMS

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Obtuse constants of Alexandrov spaces2019

    • Author(s)
      A.Mitsuishi and T. Yamaguchi
    • Journal Title

      J. Math. Soc. Japan

      Volume: 印刷中

    • NAID

      130007733314

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lipschitz Homotopy Convergence of Alexandrov Spaces,2018

    • Author(s)
      Ayato Mitsuishi and Takao Yamaguchi,
    • Journal Title

      Journal of Geometric Analysis

      Volume: 1 Issue: 3 Pages: 1-1

    • DOI

      10.1007/s12220-018-0075-6

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Short-time behavior of the heat kernel and Weyl's law on RCD^*(K, N) spaces2018

    • Author(s)
      L. Ambrosio, S. Honda and D.Tewodrose
    • Journal Title

      Ann. Global Anal. Geom.

      Volume: 53

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Continuity of nonlinear eigenvalues in CD(K, ∞) spaces with respect to measured Gromov-Hausdorff convergence2018

    • Author(s)
      L. Ambrosio, S. Honda and J. W. Portegies
    • Journal Title

      Calc. Var. Partial Differential Equations

      Volume: 57

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 固有関数による球面の特徴付け2021

    • Author(s)
      本多正平
    • Organizer
      リーマン幾何と幾何解析
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] A new sphere theorem via the eigenmap2020

    • Author(s)
      Shouhei Honda
    • Organizer
      Geometric Measure Theory and Geometric Analysis in Moscow
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A new sphere theorem via the eigenmap2020

    • Author(s)
      Shouhei Honda
    • Organizer
      2020 Virtual Workshop on Ricci and Scalar Curvature
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Ellipsoids converge to Gaussian spaces2020

    • Author(s)
      Takashi Shioya
    • Organizer
      Geometric Measure Theory and Geometric Analysis in Moscow
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Graph manifolds as ends of negatively curved Riemannian manifolds2020

    • Author(s)
      Takashi Shioya
    • Organizer
      Curvature and Topology of Spaces
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Convergence and collapsing of spaces with boundary2019

    • Author(s)
      Takao Yamaguchi
    • Organizer
      Curvature and Topology of Spaces
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Limit spaces of Riemannian manifolds with boundary2019

    • Author(s)
      Takao Yamaguchi
    • Organizer
      The 5th Japan-China Geometry Conference
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Embedding of RCD(K, N) spaces in L2 via eigenfunctions2019

    • Author(s)
      Shouhei Honda
    • Organizer
      Optimal transport and Geometric Analysis
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Non-collapsed/collapsed Gromov-Hausdorff limit spaces, synthetic treatment of Ricci curvature, and relation to complex geometry I, II2019

    • Author(s)
      Shouhei Honda
    • Organizer
      The 25th Symposium on Complex Geometry
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Collapsing Riemannian manifolds with boundary2018

    • Author(s)
      T. Yamaguchi
    • Organizer
      The 6th German-Japan University Presidents Conference
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Isoperimetric rigidityand distributions of 1-Lipschitz functions2018

    • Author(s)
      T.Shioya
    • Organizer
      The 4th Japan-China Geometry Conference
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Weyl’s law on metricsure spaces with Ricci bounds from below2018

    • Author(s)
      S. Honda
    • Organizer
      belowGeometric Analysis Seminar, Tshinghua University
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Geometric analysis on metric measure spaces with uniform Ricci bounds from below2018

    • Author(s)
      S. Honda
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential equations and Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Ricci曲率が下に有界な特異空間上の幾何解析2018

    • Author(s)
      S. Honda
    • Organizer
      018年度日本数学会幾何学賞受賞特別講演, 岡山大学
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Embedding of $RCD^*(K, N)$ spaces in $L^2$ via eigenfunctions2018

    • Author(s)
      S. Honda
    • Organizer
      Optimal Transport and Applications, Scuola Normale Superiore
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Book] Elliptic PDEs on Compact Ricci Limit Spaces and Applications2018

    • Author(s)
      Shouhei Honda
    • Total Pages
      104
    • Publisher
      AMS
    • ISBN
      9781470444174
    • Related Report
      2018 Annual Research Report

URL: 

Published: 2018-04-23   Modified: 2023-01-30  

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