• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

2005 Fiscal Year Final Research Report Summary

Convergence of metric measure spaces and energy forms

Research Project

Project/Area Number 15340053
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKanazawa University

Principal Investigator

KASUE Atsushi  Kanazawa Univ., Graduate School of Natural Science, Professor, 自然科学研究科, 教授 (40152657)

Co-Investigator(Kenkyū-buntansha) NAKAO Shintaro  Kanazawa Univ., Graduate School of Natural Science, Professor, 自然科学研究科, 教授 (90030783)
TAKANOBU Satoshi  Kanazawa Univ., Graduate School of Natural Science, Professor, 自然科学研究科, 教授 (40197124)
USHIJIMA Akira  Kanazawa Univ., Graduate School of Natural Science, Lecturer, 自然科学研究科, 講師 (50323803)
NAKAGAWA Yasuhiro  Kanazawa Univ., Graduate School of Natural Science, Assistant Professor, 自然科学研究科, 助教授 (90250662)
KATO Shin  Osaka City Univ., Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (10243354)
Project Period (FY) 2003 – 2005
KeywordsDirichlet space / spectral convergence / kernel function / energy form / Riemannian distance / netwok / effective resistance / quasi-isometry
Research Abstract

We curried out the studies on convergence of conservative regular Dirichlet spaces and some analysis of the limit spaces. The set of spaces under consideration includes particularly Riemannian manifolds, Riemannian polyhedra, sub-Riemannian manifolds. The convergence is meant by a variational convergence, called the Gamma convergence, of energy forms and spectral convergence.
The main results are described as follows :
(1)Considering a convergent sequence of open subsets of the spaces, we verified the convergence of the kernel functions, the Green functions, harmonic functions and so on. Moreover we showed a new insight into phenomena of concentration of energies of functions or more generally maps of least energy by describing it in terms of singularities of the limit spaces.
(2)We developed the convergence theory concerning metric graphs, that is, one dimensional Riemannian polyhedra. In our arguments, the effective resistance plays important roles. Using this notion, we discussed the c … More onvergence of energy forms, the Dirichlet energy forms, in a certain variatinal sense, and also the metric structure of the graphs in the Gromov-Hausdorff sense. The theory features some particular phenomena on the set of one dimensional spaces. An important class of so called fractal sets is in fact included in our limit spaces. We proposed a new approach to the analysis on fractals.
(3)Locally finite, infinite networks may be viewed as limits of finite networks. From this point of view, we studied Royden's compactification of infinite networks. Among other things, we proved the invariance of the Royden boundaries under quasi-isometric transformations, and further the metrizability of the Royden compactification under the condition of the effective resistance being bounded uniformly. Infinite networks can be considered as good approximations of complete Riemannian manifolds. We compared complete Riemannian manifolds with certain networks, focusing the functions of finite Dirichlet energies. Less

  • Research Products

    (19 results)

All 2006 2005 2004 2003 Other

All Journal Article (18 results) Book (1 results)

  • [Journal Article] Variational convergence of finite networks2006

    • Author(s)
      Atsushi Kasue
    • Journal Title

      Interdisciplinary Information Sciences 12・1

      Pages: 57-70

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Convergence of metric measure spaces and energy forms2006

    • Author(s)
      Atsushi Kasue
    • Journal Title

      Amer.Math.Soc.Translations Ser.2 215

      Pages: 79-96

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Convergence of Riemannian manifolds and Laplace operators II2006

    • Author(s)
      Atsushi Kasue
    • Journal Title

      Potential Analysis (To appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The probability of two F_q-polynomials to be coprime2006

    • Author(s)
      H.Sugita, S.Takanobu
    • Journal Title

      Advanced Studies in Pure Math. 43

      Pages: 1-24

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Variational Convergence of Finite Networks,2006

    • Author(s)
      A.Kasue
    • Journal Title

      Interdisciplinary Information Science 12/1

      Pages: 57-70

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The probability of two F_q-polynomials to be coprime,2006

    • Author(s)
      H.Sugita, S.Takanobu
    • Journal Title

      Advanced Studies in Pure Mathematics 43

      Pages: 1-24

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A volume formula for generalized hyperbolic tetrahedral2006

    • Author(s)
      A.Ushijima
    • Journal Title

      Mathematics and its Applications (New York)(Springer, New York) 581

      Pages: 249-265

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the essential spectrum of the Laplacian and vague convergence of the curvature at infinity2005

    • Author(s)
      H.Kumura
    • Journal Title

      Comm.in Partial Differential Equations 30

      Pages: 1555-1565

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Convergence of metric measure spaces and energy forms2005

    • Author(s)
      A.Kasue
    • Journal Title

      Amer.Math.Soc.Transl. (2)215

      Pages: 79-96

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the essential spectrum of the Laplacian and vague convergence of the curvatures at infinity2005

    • Author(s)
      H.Kumura
    • Journal Title

      Communications in Partial Differential Equations 30

      Pages: 1555-1565

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Geometry of an end and absence of eigenvalues in the essential spectrum2005

    • Author(s)
      H.Kumura
    • Journal Title

      math.DG 0505557(to appear)

      Pages: 26

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the weights of end-pairs in n-end catenoids of genus zero2004

    • Author(s)
      S.Kato, K.Nomura
    • Journal Title

      Osaka J.Math. 41

      Pages: 1-26

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An extension of Stieljes-Young integrals2004

    • Author(s)
      S.Nakao
    • Journal Title

      Sci.Rep.Kanazawa Univ. 48

      Pages: 1-4

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An representation of continuous additive functional of zero energy2004

    • Author(s)
      S.Nakao
    • Journal Title

      Sci.Rep.Kanazawa Univ. 48

      Pages: 5-8

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the weights of end-pairs in n-end catenoids of genus zero,2004

    • Author(s)
      S.Kato, K.Nomura
    • Journal Title

      Osaka J.Math. 41

      Pages: 1-26

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The Bando-Calabi-Futaki character and its lifting to a group character2003

    • Author(s)
      Y.Nakagawa
    • Journal Title

      Math.Ann. 325

      Pages: 31-53

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the intrinsic ultracontractivity for compact manifolds with boundary.2003

    • Author(s)
      H.Kumura
    • Journal Title

      Kyushu J.Math 57

      Pages: 29-50

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Convergence of Riemannian manifolds and Laplace operators, II,

    • Author(s)
      A.Kasue
    • Journal Title

      Potential Analysis (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 測度距離空間の幾何解析-リプシッツ関数の微分を中心に- "リーマン多様体とその極限"2004

    • Author(s)
      加須栄 篤, 大津・山口, ほか
    • Total Pages
      110
    • Publisher
      日本数学会
    • Description
      「研究成果報告書概要(和文)」より

URL: 

Published: 2007-12-13  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi