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Geometry of Harmonicity

Research Project

Project/Area Number 14340021
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

BANDO Shigetoshi  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40165064)

Co-Investigator(Kenkyū-buntansha) NISHIKAWA Seiki  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004488)
KENMOTSU Katsuei  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004404)
TAKAGI Izumi  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40154744)
URAKAWA Hajime  Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (50022679)
SUNADA Toshikazu  Meiji Univ., School of Science and Technology, Professor, 理工学部, 教授 (20022741)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥12,100,000 (Direct Cost: ¥12,100,000)
Fiscal Year 2005: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2004: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2003: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2002: ¥4,800,000 (Direct Cost: ¥4,800,000)
KeywordsFutaki invariant / complex Finsler manifold / harmonic maps / energy functional / mean curvature / reaction-diffusion system / heat kernel / crystal lattice / 活性因子・抑制因子型反応拡散系 / 概複素構造 / ヤン・ミルズ接続 / シンプレクティック多様体 / ケーラー多様体 / 解析的連接層 / 安定性 / 複素フィンスラー空間 / 回転面 / ヤング・ミルズ接続 / グラフ理論 / 概正則写像 / 完備双曲性 / CR多様体 / 平均曲線 / 活性因子-抑制因子型の反応拡散系 / ディリクレ境界値固有値問題
Research Abstract

・Bando studied the locally hyperbolically completeness of almost complex manifolds, and also showed the admissibility condition of Einstein-Hermitian metrics can be replaced by a condition which is easier to check.
・Nishikawa proposed a framework for a differential geometric proof of Hartshorne conjecture, and obtained the fundamental results. He has also conducted the differential geometric study on the foliation structures of CR-manifolds.
・Kenmotsu has extended his study of the periodicity of the surfaces of revolution with periodic mean curvature in the 3-dimensional Euclidean space to the higher dimensional case, and obtained an easier alternate proof of Hsian's result on the classification and construction of the hyper-surfaces of revolution of constant mean curvature.
・Takagi studied the dynamics of reaction-diffusion systems of activator-inhibitor type which model morphogenesis in biology, and investigated how various conditions reflect on the location of spikes in the case of dimension 1.
・Urakawa studied Yang-Mills theory and also conducted a study which relates graph theory and Riemannian geometry.
・Sunada studied the random walks on graphs as an application of the discrete geometric analysis, and established several results on periodic random walks on crystal lattices applying the large deviation theory.
・Horihata studied the initial-boundary value problem on Landau-Lifshitz-Gilbert (LLG) equation which is a model equation of magnetics, and constructed a weak solution. If the dimension is greater than 2, the weak solution converges to a constant in the infinit time provided the boundary value is a constant.

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (38 results)

All 2006 2005 2004 2003 Other

All Journal Article (21 results) Book (3 results) Publications (14 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 (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Large deviation and the tangent cone at infinity of a crystal lattice2006

    • Author(s)
      M.Kotani, T.Sunada
    • Journal Title

      Math. Z. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Large deviation and the tangent cone at infinity of a crystal lattice2006

    • Author(s)
      M.Kotani, T.Sunada
    • Journal Title

      Math.Z. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Rotational hypersurfaces of periodic mean curvature in the Euclidean spaces2005

    • Author(s)
      K.Kenmotsu
    • Journal Title

      Proceedings of the International Workshop on Integrable systems, Geometry, and Visualization, (edited by Reiko Miyaoka)

      Pages: 99-106

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Rotational hypersurfaces of periodic mean curvature in the Euclidean spaces2005

    • Author(s)
      K.Kenmotsu
    • Journal Title

      Proceedings of the International Workshop on Integrable systems, Geometry, and Visualization (edited by Reiko Miyaoka)

      Pages: 99-106

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Harmonic maps in complex Finsler geometry2004

    • Author(s)
      S.Nishikawa
    • Journal Title

      Progress in Nonlinear Differential Geometry and Their Applications, Birkhauser 59

      Pages: 113-132

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Foliated CR Manifolds2004

    • Author(s)
      S.Dragomir, S.Nishikawa
    • Journal Title

      J.Math.Soc.Japan 56

      Pages: 1031-1068

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Energy minimizer maps on C-manifolds2004

    • Author(s)
      C.Gherghe, K.Kenmotsu
    • Journal Title

      Differential Geometry and Its Applications 21

      Pages: 55-63

    • 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 Math., AMS 348

      Pages: 221-232

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

    • Author(s)
      J-S.Park, H.Urakawa
    • Journal Title

      International J.Pure Appl.Math. 10

      Pages: 79-90

    • 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] An obstruction for Chern class forms to be harmonic

    • Author(s)
      S.Bando
    • Journal Title

      Kodai Math. J. (accepted)

    • NAID

      130004948960

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Global solutions to a one-dimensional nonlinear parabolic system modeling colonial formation by chemotactic bacteria

    • Author(s)
      K.P.Htoo, M.Mimura, I.Takagi
    • Journal Title

      Asymptotic Analysis and Singularity, Advanced Studies in Pure Mathematics (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Determination of the limit sets of trajectories of the Gierer-Meinhardt system without diffusion

    • Author(s)
      W.-M.Ni, K.Suzuki, I.Takagi
    • Journal Title

      Asymptotic Analysis and Singularity, Advanced Studies in Pure Mathematics (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An obstruction for Chem class forms to be harmonic

    • Author(s)
      S.Bando
    • Journal Title

      Kodai Math.J. (accepted)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Globalsolutions to a one-dimensional nonlinear parabolic system modeling colonial formation by chemotactic bacteria

    • Author(s)
      K.P.Htoo, M.Mimura, I.Takagi
    • Journal Title

      "Asymptotic Analysis and Singularity", Advanced Studies in Pure Mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Determination of the limit sets of trajectories of the Gierer-Meinhardt system without diffusion

    • Author(s)
      W.-M.Ni, K.Suzuki, I.Takagi
    • Journal Title

      "Asymptotic Analysis and Singularity", Advanced Studies in Pure Mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Convergence rates to equilibrium of the heat kernels on compact Riemannian manifolds

    • Author(s)
      H.Urakawa
    • Journal Title

      Indiana University Mathematical Journal (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An obstruction for Chern class forms to be harmonic

    • Author(s)
      S.Bando
    • Journal Title

      Kodai Math.J. (accepted)

    • NAID

      130004948960

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Global solutions to a one-dimensional nonlinear parabolic system modeling colonial formation by chemotactic bacteria

    • Author(s)
      K.P.Htoo, M.Mimura, I.Takagi
    • Journal Title

      "Asymptotic Analysis and Singularity", Advanced Studies in Pure Mathematics (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Determination of the limit sets of trajectories of the Gierer-Meinhardt system without diffusion

    • Author(s)
      W.-M.Ni, K.Suzuki, I.Takagi
    • Journal Title

      "Asymptotic Analysis and Singularity", Advanced Studies in Pure Mathematics (to appear)

    • Related Report
      2005 Annual Research Report
  • [Book] ダイヤモンドはなぜ美しい2006

    • Author(s)
      砂田利一
    • Publisher
      シュプリンガー・フェアラーク東京(出版予定)
    • Related Report
      2005 Annual Research Report
  • [Book] Surfaces with constant mean curvature2003

    • Author(s)
      K.Kenmotsu
    • Total Pages
      142
    • Publisher
      Translations of Mathematical Monographs, vol. 221, American Mathematical Society
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Surfaces with constant mean curvature Translations of Mathematical Monographs, vol.2212003

    • Author(s)
      K.Kenmotsu
    • Total Pages
      142
    • Publisher
      American Mathematical Society
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] S.Nishikawa: "Harmonic maps in complex Finsler geometry"Geometric Variational Problems : Bubbles, Scans and Geometric Flows. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kenmotsu: "Surfaces of revolution with periodic mean curvature"Osaka Jour.Math.. 40. 687-696 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kenmotsu: "Periodic mean curvature and Bezier curves"Differential Geometry and Related Topics. 135-146 (2003)

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

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Urakawa: "Yang-Mills theory over compact symplectic manifolds"Annals Global Analysis Geometry. (to appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Sunada: "Geometric aspects of large deviations for random walks on crystal lettices"Proc.of "Microlocal Analysis and Complex Fourier Analysis". (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kenmotsu: "Surfaces with constant mean curvature"American Mathematical Society. 142 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Nishikawa: "Harmonic maps between Carnot spaces"Differential Geometry acid Related Topics. (to appear). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Dragomir, S.Nishikawa: "Foliated CR Manifolds"in the Technical Reports of the Erwin Schrodinger Institute. (to appear). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Kenmotsu: "Surfaces of revolution with periodic mean curvature"Osaka Jour. Math.. (to appear). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Kenmotsu: "Surfaces of revolution with periodic mean curvature and Bezier curves"Differential Geometry and Related Topics. (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Nagasawa, I.Takagi: "Bifurcating critical points of bending energy under constraints related to the shape of red blood cells"Calculus of Variations. 16. 63-111 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Urakawa: "The Dirichlet eigenvalue problem, the finite element method and graph theory"Contemporary Mathematics, Amer. Math. Soc.. (to appear). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Nishikawa.: "Variational Problems in Geometry"American Mathematical Society. 209 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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