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Comparison geometry of Finsler manifolds and gradient flows

Research Project

Project/Area Number 15K04844
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionOsaka University (2017-2018)
Kyoto University (2015-2016)

Principal Investigator

Ohta Shin-ichi  大阪大学, 理学研究科, 教授 (00372558)

Research Collaborator KUWADA Kazumasa  
PALFIA Miklos  
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywordsフィンスラー多様体 / リッチ曲率 / 局所化 / 等周不等式 / 凸関数 / 勾配流 / 収縮性 / 分解定理 / 曲率 / 距離空間 / 固有値 / 準凸関数 / 熱流 / 関数不等式 / フィンスラー幾何 / 最適輸送理論
Outline of Final Research Achievements

We investigated Finsler manifolds with weighted Ricci curvature bounded below, and established an isoperimetric inequality via the needle decomposition, as well as isoperimetric and functional inequalities via the nonlinear Gamma calculus. We also studied gradient flows of convex functions on metric spaces, and showed the contraction property of gradient flows in CAT(1)-spaces, and the rectifiability (finiteness of the length) of gradient curves of quasi-convex functions on CAT(0)-spaces. Furthermore, we showed that a metric measure space of positive Ricci curvature with the sharp spectral gap necessarily splits off a one-dimensional Gaussian space.

Academic Significance and Societal Importance of the Research Achievements

フィンスラー多様体は、距離の非対称性や熱流の非線形性など、従来のリーマン多様体では捉えられない現象の記述を可能にする幾何学的対象として、普遍的な価値を持つ。本研究で得られた成果はフィンスラー多様体の「曲がり方」に関する研究で強力な武器となるneedle分解とガンマ解析に基づくもので、今後の発展が見込まれる。また、凸関数の勾配流は最適化理論など工学的にも重要なものであり、本研究の成果はその理論的な理解を一歩進めるものである。

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (42 results)

All 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (9 results) Journal Article (7 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 7 results,  Open Access: 1 results,  Acknowledgement Compliant: 3 results) Presentation (20 results) (of which Int'l Joint Research: 16 results,  Invited: 16 results) Remarks (3 results) Funded Workshop (3 results)

  • [Int'l Joint Research] Steklov Institute of Mathematics(ロシア連邦)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Florence(イタリア)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Babes-Bolyai University(ルーマニア)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Duke University/Michigan State University(米国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] SISSA(イタリア)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] 成均館大学校(韓国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] SISSA(イタリア)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Freiburg(ドイツ)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Rigidity for the spectral gap on RCD(K,∞)-spaces2019

    • Author(s)
      Nicola Gigli, Christian Ketterer, Kazumasa Kuwada, Shin-ichi Ohta
    • Journal Title

      American Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Heat kernel estimates on connected sums of parabolic manifolds2018

    • Author(s)
      A. Grigor’yan, S.Ishiwata, L.Saloff-Coste
    • Journal Title

      Journal de Mathematiques Pures et Appliquee

      Volume: 113 Issue: 1 Pages: 155-194

    • DOI

      10.1007/s12220-018-00126-7

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Needle decompositions and isoperimetric inequalities in Finsler geometry2018

    • Author(s)
      Shin-ichi Ohta
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 70 Issue: 2 Pages: 651-693

    • DOI

      10.2969/jmsj/07027604

    • NAID

      130007383761

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces2017

    • Author(s)
      Shin-ichi Ohta, Miklos Palfia
    • Journal Title

      Amer. J. Math.

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Nonlinear geometric analysis on Finsler manifolds2017

    • Author(s)
      Shin-ichi Ohta
    • Journal Title

      European Journal of Mathematics

      Volume: 印刷中 Issue: 4 Pages: 916-952

    • DOI

      10.1007/s40879-017-0143-7

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Some functional inequalities on non-reversible Finsler manifolds2017

    • Author(s)
      Shin-ichi Ohta
    • Journal Title

      Proc. Indian Acad. Sci. Math. Sci.

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] (K,N)-convexity and the curvature-dimension condition for negative N2016

    • Author(s)
      Shin-ichi Ohta
    • Journal Title

      The Journal of Geometric Analysis

      Volume: 印刷中 Issue: 3 Pages: 2067-2096

    • DOI

      10.1007/s12220-015-9619-1

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Presentation] 最適輸送理論の広がり-曲がり方、熱、画像解析まで-2019

    • Author(s)
      太田慎一
    • Organizer
      自然科学の数学的原理
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] フィンスラー多様体の熱流と幾何解析2018

    • Author(s)
      太田慎一
    • Organizer
      保存則をもつ偏微分方程式の解の正則性、特異性および漸近挙動の研究
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Rectifiability of self-contracted curves with applications2018

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      The Cut Locus 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Rectifiability of self-contracted curves with applications2018

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Workshop on barycenters, convexity on metric spaces and positive operators
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Self-contracted curves in CAT(0)-spaces and their rectifiability2018

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Conference on Metrics and Measures
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Spectral gap and rigidity under positive Ricci curvature2017

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      23rd Rolf Nevanlinna Colloquium
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On weighted Ricci curvature of negative effective dimension2017

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      CMC conference: Optimal transport and related topics
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On weighted Ricci curvature of negative effective dimension2017

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Intense Activity Period "Metric Measure Spaces and Ricci Curvature"
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Rigidity for the spectral gap inequality on RCD(K,∞)-spaces2017

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Curvatures of Graphs, Simplicial Complexes and Metric Spaces
    • Place of Presentation
      三亜,中国
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Rigidity for the spectral gap inequality on RCD(K,∞)-spaces2017

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Dirichlet forms and their geometry
    • Place of Presentation
      東北大学
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Gradient flows in CAT(1)-spaces2016

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      測地線及び関連する諸問題
    • Place of Presentation
      熊本大学
    • Year and Date
      2016-01-10
    • Related Report
      2015 Research-status Report
  • [Presentation] Isoperimetric inequalities on Finsler manifolds2016

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      The Asian Mathematical Conference
    • Place of Presentation
      Bali, Indonesia
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Localization and isoperimetric inequalities2016

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      The Cut Locus: A Bridge Over Differential Geometry, Optimal Control, and Transport
    • Place of Presentation
      Bangkok, Thailand
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Nonlinear geometric analysis on Finsler manifolds: Some functional inequalities2016

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Follow-up Workshop to JTP "Optimal Transportation"
    • Place of Presentation
      Bonn, Germany
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Nonlinear geometric analysis on Finsler manifolds: Gaussian isoperimetric inequality2016

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Geometric Analysis on Riemannian and Metric Spaces
    • Place of Presentation
      京都大学
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Some functional inequalities on non-reversible Finsler manifolds2016

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Optimal Transport and Applications
    • Place of Presentation
      Pisa, Italy
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Nonlinear geometric analysis on Finsler manifolds2016

    • Author(s)
      Shin-ichi Ohta
    • 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
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Localization method and isoperimetric inequalities2015

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      確率論と幾何学
    • Place of Presentation
      東京工業大学
    • Year and Date
      2015-11-11
    • Related Report
      2015 Research-status Report
  • [Presentation] Localization and isoperimetric inequalities in Finsler geometry2015

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      第50回フィンスラー幾何学シンポジウム
    • Place of Presentation
      広島大学
    • Year and Date
      2015-10-22
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Localization method and isoperimetric inequalities2015

    • Author(s)
      Shin-ichi Ohta
    • Organizer
      Trends in Modern Geometry 2015 & 10th Pacific Rim Complex Geometry Conference
    • Place of Presentation
      東京大学
    • Year and Date
      2015-07-28
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] Web Page of Shin-ichi OHTA

    • URL

      http://www4.math.sci.osaka-u.ac.jp/~sohta/

    • Related Report
      2018 Annual Research Report
  • [Remarks] Web Page of Shin-ichi OHTA

    • URL

      http://www.math.sci.osaka-u.ac.jp/~sohta/

    • Related Report
      2017 Research-status Report
  • [Remarks] Web Page of Shin-ichi OHTA

    • URL

      https://www.math.kyoto-u.ac.jp/~sohta/

    • Related Report
      2015 Research-status Report
  • [Funded Workshop] Geometry and Probability2019

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] Geometry and Probability2018

    • Related Report
      2017 Research-status Report
  • [Funded Workshop] Geometric Analysis on Riemannian and Metric Spaces2016

    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-09-05
    • Related Report
      2016 Research-status Report

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Published: 2015-04-16   Modified: 2020-03-30  

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