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Convergence of Riemannian manifolds and Laplace operators

Research Project

Project/Area Number 12640218
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKanazawa University (2001-2002)
Osaka City University (2000)

Principal Investigator

KASUE Atsushi  Kanazawa University, Department of Mathematics, Professor, 自然科学研究科, 教授 (40152657)

Co-Investigator(Kenkyū-buntansha) KATO Shin  Osaka City University, Department of Mathematics, As. Prof., 大学院・理学研究科, 助教授 (10243354)
ISHIMOTO Hiroyasu  Kanazawa University, Department of Mathematics, Prof., 理学部, 教授 (90019472)
NAKAO Shintaro  Kanazawa University, Department of Mathematics, Prof., 理学部, 教授 (90030783)
KUMURA Hironori  Shizuoka University, Department of Mathematics, As. Prof., 理学部, 助教授 (30283336)
北原 晴夫  金沢大学, 大学院・自然科学研究科, 教授 (60007119)
橋本 義武  大阪市立大学, 理学部, 助教授 (20271182)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
KeywordsRiemannian manifolds / Laplace operators / energy forms / heat kernels / spectra / Dirichlet spaces / Gromov-Hausdorff convergence / spectral convergence
Research Abstract

Riemannian manifolds are considered as metric spaces equipped with Riemannian distances. From this point of view, a set of compact, connected Riemannian manifolds has uniform structure defined by the Gromov-Hausdorff distance, and there are intensive activities around the convergence theory of Riemannian manifolds, which include some works from the viewpoint of spectral geometry and also diffusion processes. In 1994, we introduced a spectral distance on a set of compact, (weighted) Riemannian manifolds, using heat kernels instead of Riemannian distances, and proved some results on the spectral convergence of Riemannian manifolds. In this project, we I continued the study for further developments and proved some results as follows: (1) the energy forms "dominates" the intrinsic distances in a certain sense; the relation of domination can be expressed in terms of the energy density in the limit spaces, the volume doubling property, and the scale invariant Poincare inequality, which play important roles in our theory. (2) the energy functional not only on function spaces but also on the space of maps are able to be discussed in the same vein and the convergence of the functional can be investigated in relation to the geometric and topological properties of spaces under study. (3) Riemannian vector bundles and the energy functional on them can naturally arise as the important subject of our theory. (4) Deformations of submanifolds provide new problems in our setting.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (32 results)

All Other

All Publications (32 results)

  • [Publications] A.Kasue: "Convergence of Riemannian manifolds and Laplace operators, I"Ann.I'Institut Fourier. 52・4. 1219-1257 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A.Kasue, H.Kumura: "Spectral convergence of conformally immersed surfaces with bounded mean curvature"J.Geometric Analysis. 12・4. 663-681 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Kumura: "A note on the cbsence of eigenvalues on negatively curved manifolds"Kyushu J.Math.. 56. 109-121 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Kato: "On the weight of end-pairs in n-end catenoids"Lec.Note.Ser.in Math., Osaka U.. 7. 93-108 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 加須栄 篤: "リーマンベクトル束のスペクトル収束"Lec.Note.Ser.in Math.Osaka U.. 7. 69-72 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 加須栄 篤: "測度距離空間の収束とエネルギー形式"数学(日本数学会・岩波書店). 55・1. 20-36 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. Kasue: "Convergence of Riemannian manifolds, Laplace operators and energy forms"Proceedings of the Fifth Pacific Rim Geometry Conference (ed. S. Nishikawa), Tohoku Math. Publ.. 20. 75-97 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. Kasue: "Convergence of Riemannian manifolds and Laplace operators, I"Ann. l'Institut Fourier. 52-4. 1219-1257 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. Kasue, H. Kumura: "Spectral convergence of conformally immersed surfaces with bounded mean curvature"J. Geom. Anal.. 12-4. 663-681 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. Kasue: "Convergence of measured metric spaces and energy forms (Japanese)"Sugaku Expositions, ed. Math. Soc of Japan, Iwanami. 55-1. 20-36 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. Kasue: "Spectral convergence of Riemmanian vector bundles"Lecture Notes Ser. In Mathematics 7, Osaka University. 69-92 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Kumura: "Nash inequalities for compact manifolds with boundary"Kodai Math. J.. 24. 352-378 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Kumura: "A note on the absence of eigenvalues on negatively curved manifolds"Kyushu J, Math.. 56. 109-121 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Kumura: "On the intrinsic ultracontractivity for compact manifolds with boundary"Kyushu J. Math.. to appear

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Kato, M. Umehara & K. Yamada: "General existence of minimal surfaces ofgenus zero with catenoidal ends and prescribed flux"Comm. Anal. Geom.. 8. 83-114 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Kato: "The scalar curvature equation on open Riemannian manifolds"Sugaku Expositions. 14. 219-236 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Kato: "On the weight of end-pairs in n-end catenoids (Japanese)"Lecture Note Series in Mathematics, Osaka University. 7. 93-108 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A.Kasue: "Convergence of Riemannian manifolds and Laplace operators. I"Ann. I'Institut fourier. 52・4. 1219-1257 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Kasue, H.Kumura: "Spectral convergence of conformally immersed surfaces with bounded mean curvature"J. Geometric Analysis. 12・4. 663-681 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Kumura: "A Note on the cbsence of eigenvalues on negatively curved manifolds"Kyushu J. Math.. 56. 109-121 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Kato: "On the weight of end-paivs in n-end catenoids"Lec. Note Ser. in Math, Osaka U.. 7. 93-108 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 加須栄 篤: "リーマンベクトル束のスペクトル収束"Lec. Note Ser. in Math Osaka U.. 7. 69-92 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 加須栄 篤: "測度距離空間の収束とエネルギー形式"数学(日本数学会・岩波書店). 55・1. 20-36 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Kasue: "Corvergence of Riemnnian manifolds, Laplace operators and energy forns"Proc. 5^<th> Pacific Rim Geon. Conf. Tohoku Math, Pub.. 20. 75-97 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Kasue: "Convergence of Riemannian manifolds and Laplace operators, I"Ann. Instit. Fourier. 52. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Kasue, H.Kumura: "Spectral convergence of conformally immevsed surfaces with bounded mean curvature"J. Geom. Analysis. (未定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Kumura: "Nosh inequalities faconpact Riemannian manifolds with boundary"Kodai Math. J.. 24. 352-378 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 加須栄 篤: "リーマン幾何学"培風館. 264 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Kasue: "Convergence of Riemannian manifolds and Laplace operators"Ann.Institut Fourier. (掲載予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] A.Kasue: "Convergence of Riemannian manifolds,Laplace operators and energy forms"Proc.of the Fifth Pacific Rim Geometry Conf.. (掲載予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kato: "General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux"Comm.Anal,Geom.. 8. 83-114 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 加須栄篤: "リーマン幾何学"培風館(近刊). 250 (2001)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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