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Bubbling phenomenons of harmonic maps and their compactification

Research Project

Project/Area Number 10640092
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTOHOKU UNIVERSITY (1999)
Toho University (1998)

Principal Investigator

KOTANI Makoto  Mathematical Institute, Tohoku University, Ass. Prof., 大学院・理学研究科, 助教授 (50230024)

Co-Investigator(Kenkyū-buntansha) NAKAGAWA Yasuhiro  Mathematical Institute, Tohoku University, Ass. Prof., 大学院・理学研究科, 助教授 (90250662)
IZEKI Hiroyasu  Mathematical Institute, Tohoku University, Ass. Prof., 大学院・理学研究科, 助教授 (90244409)
SUNADA Toshikazu  Mathematical Institute, Tohoku University, Prof., 大学院・理学研究科, 教授 (20022741)
MIYAOKA Reiko  Faculty of Science, Sophia University, Prof., 理工学部, 教授 (70108182)
OHNITA Yoshihiro  Faculty of Science, Tokyo Metropolitan University, Prof., 理学部, 教授 (90183764)
塚田 真  東邦大学, 理学部, 教授 (10120198)
大口 剛史  東邦大学, 理学部, 助教授 (60168888)
志村 道夫  東邦大学, 理学部, 教授 (90015868)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1999: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordscrystal lattice / harmonic map / transition probability / random walk / central limit theorem / Albanese map / 調和写線 / アルバネーゼ写線 / フロッケ理論 / 熱核 / アルバネゼ写像
Research Abstract

We discuss long time asymptotic behaviors of the heat kernel on a non-compact Riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient. A local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted Laplacians. Our ideas and techniques are motivated partly by analogy with Floque-Bloch theory on periodic Schrodinger operators. For the asymptotic expansion, we make careful use of the classical Laplace method. In the course of a discussion, we observe that the notion of Albanese maps associated with the abelian group action is closely related to the asymptotics. A similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph. The results obtained in the present paper refine our previous ones. In the asymptotics, the Euclidean distance associated with the standard realization of the lattice graph, which we call the Albanese distance, plays a crucial role.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (43 results)

All Other

All Publications (43 results)

  • [Publications] M.Kotani,T.Shirai T.Sunada: "Asymptotic behavior of the transition probability of a random walk on an infinite graph"J. Funct. Anal.. 159. 664-689 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Kotani and T.Sunada: "Jacobian tori associated with a finite graph and its abelian covering graphs"Adv. In Appl.Math.. (発売予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Kotani and T.Sunada: "Standard realization of crystal lattice via harmonic maps"Trans. Amer. Math. Soc.. (発売予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Kotani and T.Sunada: "Albanese maps and off diagonal long time asymptotics for the heat kernels"Comm. Math. Phys.. (発売予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Kotani and T.Sunada: "Zeta functions of finite graphs"J. MS. Univ. Tokyo. (発売予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Kotani and T.Sunada: "The pressure and higher correlations for an Anosov diffeomorphism"Erg. Th. Dyn. Sys.. (発売予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Sunada: "Co-growth functions and spectra of the adjacency operators for finitely generated groups"J. Math. Soc. Japan. (発売予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Izeki: "The Teichmuller distance on the space of flat conformal structures tures"Conform .Geom. Dynam.. 2. 1-24 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Izeki and S.Nayatani: "Canonical metric on the domain of discontinuity of a Kleinian group"Semin. Theor. Spectre. Geom. Univ. Grenoble I. 16. 9-32 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Nakagawa: "Bando-Calabi-Futaki characters of Kahler orbifolds"Math. Ann.. 314. 369-380 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R.Miyaoka: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces"Tohoku Math. J.. 51. 35-53 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R.Miyaoka,R.Aiyama,K.Akutagawa,M.Umehara: "A global correspondence between CMC-surfaces in P^3(R) and pairs of non-holomorphic harmonic maps into S^2"Proceedings of Amer. Math. Soc.. 128. 939-941 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Kotani, T. Shirai and T. Sunada: "Asymptotic behavior of the transition probability of a random walk on an infinite graph"J. Funct. Anal. 159. 664-689 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Kotani and T. Sunada: "Jacobian tori associated with a finite graph and its abelian covering graphs"Adv. in Appl. Math.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Kotani and T. Sunada: "Standard realization of crystal lattice via harmonic maps"Trans. Amer. Math. Soc.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Kotani and T. Sunada: "Albanese maps and off diagonal long time asymptotics for the heat kernels"Comm. Math. Phys.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Kotani and T. Sunada: "Zeta functions of finite graphs"J. MS. Univ. Tokyo. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Kotani and T. Sunada: "The pressure and higher correlations for an Anosov diffeomorphism"Erg. Th. Dyn. Sys.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Sunada: "Co-growth functions and spectra of the adjacency operators for finitely generated groups"J. Math. Soc. Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Sunada: "Twisted group C* algebras applied to spectral geometry of noncompact manifolds"Surveys in Geometry. (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Izeki: "The Teichmuller distance on the space of flat conformal structures"Conform. Geom. Dynam.. 2. 1-24 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Izeki and S. Nayatani: "Canonical metric on the domain of discontinuity of a Kleinian group"Semin. Theor. Spectr. Geom., Univ. Grenoble I. 16. 9-32 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nakagawa: "Bando-Calabi-Futaki characters of Kahler orbifolds"Math. Ann.. 314. 369-380 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Miyaoka: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces"Tohoku Math. J.. 51. 35-53 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Miyaoka: "Hypersurface geometry and Hamiltonnian systems of hydrodynamic type"Lobachevskii J.. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Miyaoka, R. Aiyama, K. Akutagawa, and M. Umehara: "A global correspondence between CMC-surfaces in PィイD13ィエD1(R) and pa irs of non-holomorphic harmonic maps into SィイD12ィエD1"Proceedings of Amer. Math. Soc.. (to appear). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A. Tachikawa: "Weak Solution to Evolution Problems of Harmonic Maps from Noncompact Manifolds"Rendiconti di Matematica. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Kotani,T.Shirai,T.Sunada: "Asymptotic behavior of the transition probability of a random walk on an infinite graph"J. Funct. Anal.. 159. 664-689 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Kotani and T.Sunada: "Jacobian tori associated with a finite graph and its abelian covering graphs"Adv. in Appl. Math.. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Kotani and T.Sunada: "Standard realization of crystal lattice via harmonic maps"Trans. Amer. Math. Soc.. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Kotani and T.Sunada: "Albanese maps and off diagonal long time asymptotics for the heat kernels"Comm. Math. Phys.. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Kotani and T.Sunada: "Zeta functions of finite graphs"J. MS. Univ. Tokyo. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Kotani and T.Sunada: "The pressure and higher correlations for an Anosov diffeomorphism"Erg.Th. Dyn. Sys.. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Sunada: "Co-growth functions and spectra of the adiacency operators for finitely generated groups"J. Math. Soc. Japan. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Izeki: "The Teichmuller distance on the space of flat conformal structures"Conform. Geom. Dynam. 2. 1-24 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Izeki and S.Nayatani: "Canonical metric on the domain of discontinuity of a Kleinian group"Semin. theor. spectr. geom., Univ. Grenoble I. 16. 9-32 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Nakagawa: "Bando-Calabi-Futaki characters of Kahler orbifolds"Math. Ann.. 314. 369-380 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Miyaoka: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces"Tohoku Math. J.. 51. 35-53 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Miyaoka,R.Aiyama,Akutagawa,M.Umehara: "A global correspondence between CMC-surfaces in P^3(R) and pairs of non-holomorphic harmonic maps into S^2"Proceedings of Amer. Math. Soc.. 128. 939-941 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Kotani, T.Shirai, T.Sinoda: "Asympistic behavior of the transitun proba orlity of a romdom walk on and infirto graph" J.Functional Analysis. 159. 664-689 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Kotani, T.Sinoda: "Jocobion tori associated with a fnite gropl and its clelian coverig graphs" Advance in Applied Mathematics.

    • Related Report
      1998 Annual Research Report
  • [Publications] R.Miyaoka: "The splitting and obformetions of the generchzed Gahzmap of copaet cihc surbacis" Tohoku.Math.Jonal. 51. 35-53 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Ohnita M.Guest: "Acturno of loop aporps,difonntcrs of Rennoric mops and the aprlnutiro" Sugcku Exposition(Ame.Math.Soc.). 183. 33-50 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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