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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)
有澤 真理子  東北大学, 大学院・情報科学研究科, 助教授 (50312632)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥16,300,000 (Direct Cost: ¥16,300,000)
Fiscal Year 2006: ¥5,200,000 (Direct Cost: ¥5,200,000)
Fiscal Year 2005: ¥5,000,000 (Direct Cost: ¥5,000,000)
Fiscal Year 2004: ¥6,100,000 (Direct Cost: ¥6,100,000)
Keywordsheat kernel / convergence rate / Ricci curvature / Yang-Mills connection / bi-Yang-Mills connection / biharmonic map / Dirichlet eigenvalue problem / visualization / ディリクレ境界値固有値問題 / ノイマン境界値固有値問題 / 数値計算プログラム / 調和写像 / ヤング・ミルズ場 / 2-ヤング・ミルズ場 / 孤立現象 / サイバーグ・ウィッテン方程式 / 複素スピン構造 / ディラック作用素 / ケーラー多様体 / カラビ・ヤウ多様体 / ノイマン固有値問題 / コーシー・リーマン・リー群 / レビ平坦構造
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.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (36 results)

All 2007 2006 2005 2004

All Journal Article (31 results) Book (3 results) Patent(Industrial Property Rights) (2 results)

  • [Journal Article] Visualization of the eigenvalue problems of the Laplacian for embedded surfaces and its applications2007

    • Author(s)
      H.Urakawa
    • Journal Title

      The Proceedings of Socie Math. de France (印刷決定)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Conformally flat hypersurfaces with cyclic Guichard net2007

    • Author(s)
      Y.Suyama
    • Journal Title

      International J. Math. (発表受理印刷中)

    • Related Report
      2006 Annual Research Report
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Yang-Mills fields on CR manifolds2006

    • Author(s)
      H.Urakawa
    • Journal Title

      Journal of Mathematical Physics 47

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

    • Author(s)
      H.Urakawa
    • Journal Title

      Journal of Mathematical Physics Vol. 47

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [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(1)

      Pages: 259-288

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Yang-Mills field on CR manifolds2006

    • Author(s)
      H.Urakawa
    • Journal Title

      Journal of Mathematical Physics 47(8)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian2006

    • Author(s)
      J.Takahashi
    • Journal Title

      Manuscripta Math. 121

      Pages: 191-200

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Some restrictions on weight enumerators of singly even self-dual codes2006

    • Author(s)
      A.Munemasa
    • Journal Title

      IEEE Trans. Inform. Theory 52

      Pages: 1266-1269

    • NAID

      120002337725

    • Related Report
      2006 Annual Research Report
  • [Journal Article] 朝倉書店2006

    • Author(s)
      浦川 肇
    • Journal Title

      微積分の基礎

      Pages: 232-232

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Convergence rates to equilibrium of the heat kernels on compact Riemannian manifolds2006

    • Author(s)
      H. Urakawa
    • Journal Title

      Indiana University Mathematical Journal (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 高次元ケーラー多様体上のサイバーグ・ウィッテン理論2006

    • Author(s)
      浦川 肇
    • Journal Title

      研究集会「大阪大学微分幾何研究集会」報告集

      Pages: 95-99

    • Related Report
      2005 Annual Research Report
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Yang-Mills fields analogue of biharmonic maps2005

    • Author(s)
      H. Urakawa
    • Journal Title

      Topics Hermitian Geometry and Related Fields

      Pages: 41-49

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Visualization of the eigenvalues and eigenfunctions for the Laplacian on compact embedded surfaces with or without boundaries2005

    • Author(s)
      H. Urakawa
    • Journal Title

      Proceedings of International Workshop on Integrable Systems Geometry and Visualization

      Pages: 29-79

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Total curvature of noncompact piecewise Riemannian 2-polyhedra2005

    • Author(s)
      J. Itoh
    • Journal Title

      Tsukuba Journal of Mathematics 29

      Pages: 471-493

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the length of the cutlocus for finitely many points2005

    • Author(s)
      J. Itoh
    • Journal Title

      Advances in Geometry 5

      Pages: 97-106

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 数学と工学との出会い--連続系を中心に--2005

    • Author(s)
      浦川 肇
    • Journal Title

      数学の楽しみ 冬号

      Pages: 122-137

    • Related Report
      2004 Annual Research Report
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The heat kernel and Green kernel of an infinite graph2004

    • Author(s)
      H.Urakawa
    • Journal Title

      Contemporary Mathematics, Amer.Math.Soc. 347

      Pages: 245-258

    • Related Report
      2004 Annual Research Report
  • [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

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Yang-Mills theory over compact symplectic manifolds2004

    • Author(s)
      H.Urakawa
    • Journal Title

      Annals Global Analysis Geometry 25

      Pages: 365-402

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Yang-Mills connections in homogeneous principal fiber bundles2004

    • Author(s)
      H.Urakawa
    • Journal Title

      International J.Pure Applied Mathematics 10

      Pages: 79-90

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 数学と工学との出会い--離散系を中心に--2004

    • Author(s)
      浦川 肇
    • Journal Title

      数学通信 第9巻 No.3

      Pages: 5-17

    • NAID

      40006493219

    • Related Report
      2004 Annual Research Report
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [Book] 数理システム科学2005

    • Author(s)
      熊原啓作, 編
    • Total Pages
      338
    • Publisher
      日本放送出版協会
    • Related Report
      2005 Annual Research Report
  • [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
    • Filing Date
      2005-05-06
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [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)
      浦川肇、十文宇正樹
    • Industrial Property Rights Holder
      浦川肇、十文宇正樹
    • Industrial Property Number
      2005-134794
    • Filing Date
      2005-05-06
    • Related Report
      2005 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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