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2006 Fiscal Year Final Research Report Summary

Global analysis of the heat kernels on Riemannian manifolds and graphs

Research Project

Project/Area Number 16340044
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

URAKAWA Hajime  Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (50022679)

Co-Investigator(Kenkyū-buntansha) ASOH Tohl  Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院情報科学研究科, 助教授 (00111352)
MUNEMASA Akihiro  Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (50219862)
KANEKO Makoto  Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (10007172)
OBATA Nobuaki  Tohoku University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (10169360)
ITOH Jin-ichi  Kumamoto University, Faculty of Education, Professor, 教育学部, 教授 (20193493)
Project Period (FY) 2004 – 2006
Keywordsheat kernel / convergence rate / Ricci curvature / Yang-Mills connection / bi-Yang-Mills connection / biharmonic map / Dirichlet eigenvalue problem / visualization
Research Abstract

The heat kernels of compact Riemannian manifolds converge to the equilibrium when time goes to infinity. We studied the rates of the heat kernels how do they reflect from the geometric structures of Riemannian manifolds. We showed the convergence rates are Lipshitz continuous on the deformation of Riemannian manifolds, we gave their upper estimation in terms of Ricci curvature and diameter, and also the upper estimation in terms of the non-zero first eigenvalue of the Laplacian. We gave their precise lower and upper estimations in the case of compact Riemannian symmetric spaces of rank one.
A Yang-Mills connection is a critical point of the Yang-Mills functional, and this is an analogue of harmonic map which is a critical point of the energy functional. Recently, the notion of biharmonic map was introduced which is a critical point of the 2-energy functional. We introduced the notion of 2-Yang-Mills connection which is a critical point of the 2-Yang-Mills functional. This notion is a natural generalization of Yang-Mills connection, and many further studies would be expected.
We introduced quite new method to visualize the Dirichlet or Neumann boundary eigenvalue problem of the Laplacian on plane domains. This method improved 20 percents fast comparing the known methods and reduced many steps input the data into computers. This new method made visualizations of the eigenvalue problems of the Laplacian on compact surfaces and bounded three dimensional domains. We applied to get patent of this method for programming of computer.

  • Research Products

    (15 results)

All 2006 2005 2004

All Journal Article (12 results) Book (2 results) Patent(Industrial Property Rights) (1 results)

  • [Journal Article] Convergence rates to equilibrium of the heat kernels on compact Riemannian manifolds2006

    • Author(s)
      H.Urakawa
    • Journal Title

      Indiana University Mathematical Journal 55

      Pages: 259-288

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Yang-Mills fields on CR manifolds2006

    • Author(s)
      H.Urakawa
    • Journal Title

      Journal of Mathematical Physics 47

      Pages: 08504, 1-41

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Convergence rates to equilibrium of the heat Kernels on compact Riemannian manifolds.2006

    • Author(s)
      H.Urakawa
    • Journal Title

      Indiana University Mathematical Journal Vol. 55

      Pages: 259-288

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Yang-Mills fields on CR manifolds.2006

    • Author(s)
      H.Urakawa
    • Journal Title

      Journal of Mathematical Physics Vol. 47

      Pages: 08504.1-41

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Yang-Mills fields analogue of biharmonic maps2005

    • Author(s)
      H.Urakawa
    • Journal Title

      Topics in almost Hermitian geometry and related fields

      Pages: 41-49

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Yang-Mills fields analogue of biharmonic maps.2005

    • Author(s)
      H.Urakawa
    • Journal Title

      Topics in almost Hermitian geometry and related fields

      Pages: 41-49

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The heat kernel and the Green kernel of an infinite graph2004

    • Author(s)
      H.Urakawa
    • Journal Title

      Contemporary Mathematics, Amer. Math. Soc. 347

      Pages: 245-258

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The Dirichlet eigenvalue problem, the finite element method and graph theory2004

    • Author(s)
      H.Urakawa
    • Journal Title

      Contemporary Mathematics, Amer. Math. Soc. 348

      Pages: 221-232

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Yang-Mills theory over compact symplectic manifolds2004

    • Author(s)
      H.Urakawa
    • Journal Title

      Annals of Global Analysis and Geometry 25

      Pages: 365-402

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The heat kernel and the Green kernel of an infii graph.2004

    • Author(s)
      H.Urakawa
    • Journal Title

      Contemporary Mathematics, Amer.Math.Soc. Vol.347

      Pages: 245-258

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The Dirichlet eigenvalue problem, the finite element method and graph theory.2004

    • Author(s)
      H.Urakawa
    • Journal Title

      Contemporary Mathematics, Amer.Math.Soc. Vol.348

      Pages: 221-232

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Yang-Mills theory over compact symplectic manifolds.2004

    • Author(s)
      H.Urakawa
    • Journal Title

      Annals of Global Analysis and Geometry Vol.25

      Pages: 365-402

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 微積分の基礎2006

    • Author(s)
      浦川 肇
    • Total Pages
      232
    • Publisher
      朝倉書店
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] Foundation of Calculus2006

    • Author(s)
      H.Urakawa
    • Total Pages
      232
    • Publisher
      Asakura Shoten
    • Description
      「研究成果報告書概要(欧文)」より
  • [Patent(Industrial Property Rights)] 有限要素法直接計算プログラムおよび解析方法2005

    • Inventor(s)
      甫川肇, 十文字正樹
    • Industrial Property Rights Holder
      東北大学
    • Industrial Property Number
      待願2005-134797
    • Filing Date
      2005-05-06
    • Description
      「研究成果報告書概要(和文)」より

URL: 

Published: 2008-05-27  

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