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Global Analysis of the heat kernel and Green kernel of an Infinite Graph

Research Project

Project/Area Number 13440051
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) ARISAWA Mariko  Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院・情報科学研究科, 助教授 (50312632)
KANEKO Makoto  Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (10007172)
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)
OBATA Nobuaki  Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (10169360)
OHNO Yoshiki  Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院・情報科学研究科, 助教授 (80005777)
内田 興二  東北大学, 大学院・情報科学研究科, 教授 (20004294)
伊藤 仁一  熊本大学, 教育学部, 助教授 (20193493)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥14,700,000 (Direct Cost: ¥14,700,000)
Fiscal Year 2003: ¥4,700,000 (Direct Cost: ¥4,700,000)
Fiscal Year 2002: ¥4,500,000 (Direct Cost: ¥4,500,000)
Fiscal Year 2001: ¥5,500,000 (Direct Cost: ¥5,500,000)
KeywordsYang-Mills connection / Dirichlet eigenvalue problem / finite element method / infinite graph / affine connection / oseudharmonic map / Green kernel / symplectic manifold / ヤング・ミズル接続 / ワイル接続 / モデュライ空間 / チーガー定数 / 熱核 / 鋭角三角形分割 / 境界値問題 / 固有値 / 離散計算幾何 / 非退化CR多様体 / 擬エネルギー汎関数 / 共役接続 / ヤングミルズ接続
Research Abstract

We have obtained the following results:
(1)We constructed the theory of Yang-Mills connections over compact symplectic manifolds.
(2)We estimated the Cheeger constant, the heat kernel and the Green kernel for an infinite graph in terms of the volume growth, growth of in and out degree.
(3)We determined the stiffness and mass matrices of the finite element method for the Dirichlet eigenvalue problem for a plane domain.
(4)We calculated the Cheeger constant, the heat kernel and Green kernel of semi-regular infinite graphs and gave the explicit comparison theorem for every infinite graph.
(5)We extended Yang-Mills theory to Weyl structure, and established Atiya-Hitchin-Singer theory to Weyl manifolds, and to affine connections.
(6)We formulated discrete improper affine surface theory and show its loop group description.
(7)We showed the relation in affine differential geometry, Weyl geometry, Yang-Mills theory.
(8)We defined the notion of pseudoharmonic maps from CR manifolds to a Riemanninan manifold, and showed the first variation formula and the second variation formula.
(9)We clarified the relation of each Yang-Mills theory on Kaehler manifolds, CR manifolds, and symplectic manifolds, and characterized the minimizers of the Yang-Mills functional over compact symplectic manifolds.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (32 results)

All Other

All Publications (32 results)

  • [Publications] H.Urakawa: "Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds"Indiana University Mathmatical Journal. 50. 719-745 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Ohtsuka: "Total excess on length surface"Mathematische Annalen. 319. 675-706 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Itoh: "A Sard theorem for the distance function to the cut locus"Transaction Amer.Math.Soc.. 353. 21-40 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Urakawa: "Discrete improper affine spheres"Journal Geometry and Physics. 45. 164-183 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Urakawa: "Yang-Mills theory and conjugate connections"Differential Geometry and Its Applications. 18. 229-238 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Obata: "Central limit theorems for large graphs : Method of quantum decomposition"Journal Mathematical Physics. 44. 71-88 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Urakawa: "Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds"Indiana University Mathematical Journal. 50. 719-746 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Ohtsuka: "Total excess on length surfaces"Math.Ann.. 319. 675-706 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Itoh: "The Lipschitz continuity of the distance function to the cut locus"Trans.Amer.Math.Soc.. 353. 21-40 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Urakawa: "Discrete improper affine spheres"Journal Geometry Physics. 45. 164-183 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Urakawa: "Yang-Mills theory and conjugate connections"Differential Geometry Its Applications. 18. 229-238 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Obata: "Central limit theorems for large graphs Method of quantum decomposition"Journal of Mathematical Physics. 44. 71-88 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Urakawa: "Yang-Mills theory over compact symplectic manifolds"Annals Global Analysis Geometry. (印刷中). 1-38 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Urakawa: "The heat kernel and the Green kernel of an infinite graph"Contemporary Mathematics, Amer.Math.Soc.. (印刷中). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Urakawa: "The Dirichlet eigenvalue problem, the finite element method and graph theory"Contemporary Mathematics, Amer.Math.Soc.. (印刷中). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Urakawa: "The Cheeger constant, the heat kernel and the Green kernel of an infinite graph"Monatshefte fur Mathematics. 138. 225-237 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Urakawa: "Yang-Mills theory and conjugate connections"Differential Geometry and Its Applications. 18. 229-238 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Urakawa: "Discrete improper affine spheres"J.Geometry and Physics. 45. 164-183 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Kumura: "Spectral convergence of conformally immersed surfaces with bounded mean curvature"J.Geometric Analysis. 12. 663-681 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Itoh: "Acute triangulations of the regular icosahedral surface"Discrete Computational Geometry. (発表予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Itoh: "Voronoi diagram in simply conneted complete manifold"IEICE Trans.Fundamentals. E85-A. 944-948 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Katsuda: "Stability and reconstruction in Gelfand inverse boundary spectral problem"Lecture Notes in Math., Springer. (発表予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Urakawa: "The Cheeger constant, the heat pernel and the Green bernel of an infinite graph"Monatshefte fur Mathematik. (発表予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 浦川肇: "スペクトル幾何学とグラフ理論"応用数理. 12. 29-45 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 浦川肇: "数理システム科学"日本放送協会出版会. 230 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Urakawa: "Pseudoharmonic mays from nondegenerate CR manifolds to Riemannion manifolds"Indian University Mathematics Journal. 50. 719-746 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Urakawa: "Yang-Mills theory and conjugate connections"Differential Geometry and Its Applications. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 浦川 肇: "スペクトル幾何学とグラフ理論"応用数理. 12巻1号. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.kumura: "Nash inequalities for compact manifolds with foundary"Kodoi Mathematics T.. 24. 352-378 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Itoh: "A Sard theorem for the distance function"Mathematioche Annalen. 320. 1-10 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] F.Ohtsuka: "Total excess on length surfaces"Mathematische Annalen. 319. 675-706 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 浦川 肇: "数理システム科学「連続システム」"放送大学. 72 (2002)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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