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Gromov-Hausdorff convergence and a theory of variational convergences

Research Project

Project/Area Number 17540058
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

SHIOYA Takashi  Tohoku University, Tohoku University, Graduate School of Science, Professor (90235507)

Co-Investigator(Kenkyū-buntansha) KUWAE Kazuhiro  Faculty of Education, Kumamoto University, Associate Professor (80243814)
FUJIWARA Koji  Graduate School of Information Sciences, 大学院・情報科学研究科, Professor (60229078)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsGeometry / Analysis / アレクサンドロフ空間 / Bishop-Gromov条件 / リッチ曲率 / ラプラシアン / 有界変動関数 / グリーンの公式 / 分割定理 / 比較定理 / エネルギー汎関数 / 測度距離空間
Research Abstract

In these days, the study of geometric analysis on metric measure spaces is going around. The head investigator, Shioya, Studies such a subject and his main interest is curvature of metric measure spaces and convergence, especially Alexandrov spaces, Ricci curvature of metric measure spaces, and Gromov-Hausdorff convergence of metric measure spaces. On he other hand, Mosco studied variational convergences, which is a functional analytic theory of convergence of Dirichlet energy forms. We, Shioya and the investigator, Kuwae, thought that Mosco's theory is deeply related with the study of convergence of metric measure spaces, and have extended the theory in the geometric viewpoint. We have completed it in the period of this project. The concept of convergence in our theory is nowadays called the Mosco-Kuwae-Shioya convergence and is being widely applied to the finite dimensional method in probability theory and also to some homogenization problems.
Another study is on a Laplacian comparison theorem and a splitting theorem on Alexandrov spaces with some condition corresponding to a lower bound of Ricci curvature. This is still on going. For Riemannian manifods, the Ricci curvature being bounded below is equivalent to an infinitesimal version of the Bishop-Gromov inequality. Since it is impossible to define the Ricci curvature tensor on Alexandrov spaces, we consider the infinitesimal Bishop-Gromov inequality instead of the Ricci curvature bound. Different from Riemannian, the cut-locus is not necessarily a closed set in an Alexandrov space. That may even be a dense set. By this reason, the same proof as for Riemannian manifolds does not work and we develop a new method of proof.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (38 results)

All 2008 2007 2006 2005 Other

All Journal Article (25 results) (of which Peer Reviewed: 10 results) Presentation (13 results)

  • [Journal Article] Variational convergence over metric spaces2008

    • Author(s)
      Kuwae, Kazuhiro; Shioya, Takashi
    • Journal Title

      Trans. Amer. Math. Soc. 360

      Pages: 35-75

    • NAID

      120003780146

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Variational convergence over metric spaces2008

    • Author(s)
      K. Kuwae, T. Shioya
    • Journal Title

      Trans. Amer. Math. Soc. 360

      Pages: 35-75

    • NAID

      120003780146

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Quasi-homomorphisms on mapping class groups2007

    • Author(s)
      Bestvina, Mladen; Fujiwara, Koji
    • Journal Title

      Glas. Mat. Ser. III 42(62)

      Pages: 213-236

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A note on spaces of asymptotic dimension one2007

    • Author(s)
      Fujiwara, Koji; Whyte, Kevin
    • Journal Title

      Algebr. Geom. Topol. 7

      Pages: 1063-1070

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Kato class measures of symmetric Markov processes under heat kernel estimates2007

    • Author(s)
      Kuwae, Kazuhiro; Takahashi, Masayuki
    • Journal Title

      J. Funct. Anal. 250

      Pages: 86-113

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Quasi-homomorphisms on mapping class groups2007

    • Author(s)
      Bestvina, Mladen: Fujiwara, Koji
    • Journal Title

      Glas. Mat. Ser. III 42( 62)

      Pages: 213-236

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A note on spaces of asymptotic dimension one2007

    • Author(s)
      Fujiwara, Koji; Whyte, Kevin
    • Journal Title

      Algebr. Geom. Topol 7

      Pages: 1063-1070

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Kato class measures of symmetric Markov processes under heat kernel estimates2007

    • Author(s)
      Kuwae, Kazuhiro; Takahashi, Masayuki
    • Journal Title

      J. Funct. Anal 250

      Pages: 86-113

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A note on spaces of asymptotic dimension one2007

    • Author(s)
      K. Fujiwara, K. Whyte
    • Journal Title

      Algebra. Geom. Top. 7

      Pages: 1063-1070

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Kato class measures of symmetric Markov processes under heat kernelestimates2007

    • Author(s)
      K. Kuwae, M. Takahashi
    • Journal Title

      J. Funct. Anal. 250

      Pages: 86-113

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Fixed point sets of parabolic isometries of CAT(0)-spaces2006

    • Author(s)
      Fujiwara, Koji; Shioya, Takashi; Koichi Nagano
    • Journal Title

      Comment. Math. Helv. 81

      Pages: 305-335

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Fixed point sets of parabolic isometries of CAT(O)-spaces2006

    • Author(s)
      Fujiwara, Koji; Shioya, Takashi; Koichi Nagano
    • Journal Title

      Comment. Math. Helv. 81

      Pages: 305-335

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] JSJ-decompositions of finitely presented groups and complexes of groups2006

    • Author(s)
      K Fujiwara, P. papasoglu
    • Journal Title

      Geom. Funct. Anal. 16・1

      Pages: 70-125

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Fixed point sets of parabolic isometries of CAT(o)-spaces2006

    • Author(s)
      K.fujiwara, K.nagano, T.Shioya
    • Journal Title

      Comment. Math. Helv. 81・2

      Pages: 305-335

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Kato class functions of Markov processes under ultracontractivity2006

    • Author(s)
      K.kuwae, M.Takahashi
    • Journal Title

      Potential theory in Matsue, Adr. Stad. Pure Math. 44

      Pages: 193-202

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On non-bounded generation of discrete subgroups in rank-1 Lie group2005

    • Author(s)
      Fujiwara, Koji
    • Journal Title

      Contemp. Math. 387

      Pages: 153-156

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Volume collapsed three-manifolds with a lower curvature bound2005

    • Author(s)
      Yamaguchi, Takao; Shioya, Takashi
    • Journal Title

      Math. Ann. 333

      Pages: 131-155

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On non-bounded generation of discrete subgroups in rank-1 Lie group2005

    • Author(s)
      Fujiwara, Koji
    • Journal Title

      Contemp. Math 387

      Pages: 153-156

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Volume collapsed three-manifolds with a lower curvature bound2005

    • Author(s)
      Yamaguchi, Takao; Shioya, Takasbi
    • Journal Title

      Math. Ann 333

      Pages: 131-155

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Volume collapsed three-manifolds with a lower curvature bound2005

    • Author(s)
      T.Yamaguchi, T.Shioya
    • Journal Title

      Math.Ann. 333・1

      Pages: 131-155

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Variational convergence over metric spaces

    • Author(s)
      K.kuwae, T.Shioya
    • Journal Title

      Trans. Amer. Math. Soc. (掲載決定)(to appear in)

    • NAID

      120003780146

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Variational convergence over metric spaces

    • Author(s)
      K.Kuwae, T.Shioya
    • Journal Title

      Trans.of Amer.Math.Soc. (掲載予定)(未定)

    • NAID

      120003780146

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Fixed point sets of parabolic isometries of CAT(0)-spaces.

    • Author(s)
      K.Fujiwara, T.Shioya, K.Nagano
    • Journal Title

      Comment.Math.Helv. (掲載予定)(未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Maximal principles for subharmonic functions via local semi-Dirichlet forms

    • Author(s)
      K.Kuwae
    • Journal Title

      Canadian J.Math. (掲載予定)(未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] JSS-decompositions of finitely presented groups and complexes of groups

    • Author(s)
      K.Fujiwara, P.Papasoglu
    • Journal Title

      Geom.Funct.Anal. (掲載予定)(未定)

    • Related Report
      2005 Annual Research Report
  • [Presentation] Laplacian on Alexandrov spaces2007

    • Author(s)
      Shioya, Takashi
    • Organizer
      Stochastic Calculus on Manifolds, Graphs, and Rondom Structures
    • Place of Presentation
      Hausdorff Research Institute for Mathematics, Bonn, Germany
    • Year and Date
      2007-10-09
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Laplacian on Alexandrov spaces2007

    • Author(s)
      Shioya, Takashi
    • Organizer
      Riemannian Geometry and Geometric Analysis
    • Place of Presentation
      Tsukuba University
    • Year and Date
      2007-03-06
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Concentration of measure phenomenon2007

    • Author(s)
      Shioya, Takashi
    • Organizer
      Lectures
    • Place of Presentation
      Tsukuba University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Laplacian on Alexandrov spaces2006

    • Author(s)
      Shioya, Takashi
    • Organizer
      The Second Japan-China Conference on Differential Geometry
    • Place of Presentation
      Kunmin, China
    • Year and Date
      2006-12-15
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Collapsing three-manifolds2006

    • Author(s)
      Shioya, Takashi
    • Organizer
      Colloquium
    • Place of Presentation
      Tohoku University
    • Year and Date
      2006-12-11
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Collapsing three-manifolds2006

    • Author(s)
      Shioya, Takashi
    • Organizer
      Colloquium
    • Place of Presentation
      Tokai University
    • Year and Date
      2006-12-06
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Looking at curved spaces---From an introduction to geometry to the Poincare conjecture---2006

    • Author(s)
      Shioya, Takashi
    • Organizer
      Lecture Meeting of Mathematical Culture
    • Place of Presentation
      Tohoku University
    • Year and Date
      2006-10-14
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Collapsing three-manifolds2006

    • Author(s)
      Shioya, Takashi
    • Organizer
      Annual Meeting of Mathematical Society of Japan
    • Place of Presentation
      Ohsaka City University
    • Year and Date
      2006-09-19
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Laplacian on Alexandrov spaces2006

    • Author(s)
      Shioya, Takashi
    • Organizer
      Geometry Symposium
    • Place of Presentation
      Kanazawa University
    • Year and Date
      2006-08-05
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Laplacian comparison and a splitting theorem for Alexandrov spaces2005

    • Author(s)
      Shioya, Takashi
    • Organizer
      Differential Geometry Conference at Saga University
    • Place of Presentation
      Saga University
    • Year and Date
      2005-12-17
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Laplacian on Alexandrov spaces2005

    • Author(s)
      Shioya, Takashi
    • Organizer
      Stochastic Seminar
    • Place of Presentation
      University of Bonn, Germany
    • Year and Date
      2005-12-05
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Variational convergence over metric spaces2005

    • Author(s)
      Shioya, Takashi
    • Organizer
      Heat Kernels, Stochastic Processes and Functional Inequalities
    • Place of Presentation
      MathematischesForschungsinstitut Oberwolfach, Germany
    • Year and Date
      2005-12-01
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Geometric variational convergence2005

    • Author(s)
      Shioya, Takashi
    • Organizer
      Geometry Seminar
    • Place of Presentation
      Okayama University of Science
    • Year and Date
      2005-11-11
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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