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Harnack's Inequality in Riemannian Geometry

Research Project

Project/Area Number 09440040
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOsaka City University, Department of Methematics

Principal Investigator

KASUE Atsushi  Osaka City Univ.,Dep. Of Math., Prof, 理学部, 教授 (40152657)

Co-Investigator(Kenkyū-buntansha) HASHIMOTO Yoshitake  Osaka City Univ., Dep. of Math., Assoc. Prof., 理学部, 講師 (20271182)
NISHIO Masaharu  Osaka City Univ., Dep. of Math., Assoc. Prof., 理学部, 助教授 (90228156)
KATO Shin  Osaka City Univ., Dep. of Math., Assoc. Prof., 理学部, 助教授 (10243354)
KUMURA Hironori  Shizuoka Univ., Dep. of Math., Assit, 理学部, 助手 (30283336)
OGURA Yukio  Saga Univ., Dep. of Math., Prof., 理工学部, 教授 (00037847)
KOMATSU Takashi  Osaka City Univ., Dep. of Math., Prof.
MASUDA Mikiya  Osaka City Univ., Dep. of Math., Prof. (00143371)
小松 孝  大阪市立大学, 理学部, 教授 (80047365)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥5,500,000 (Direct Cost: ¥5,500,000)
Fiscal Year 1999: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1998: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1997: ¥1,900,000 (Direct Cost: ¥1,900,000)
KeywordsRiemannian manifold / heat kernel / Gromov-Hausdorff distance / spectral distance / Dirichlet space / parabolic Harnack inequality / スペクトル距離 / 放物的ハルナック不等式 / エネルギー形式 / ハウスドルフ収束 / ガンマ収束 / リッチ曲率 / ハルナックの不等式 / ガウス型評価 / リー群の作用
Research Abstract

Riemannian manifolds are considered as metric spaces equipped with Riemannian distances and also Dirichlet spaces endowed with the Riemannian measures and the energy forms ; on a family of such manifolds, the Gromov-Hausdorff distance and the spectral distance induce uniform topologies. The former is concerning the metric structure and the latter is concerning the spectral structure.
In this program, we studied a family of compact, connected Riemannian manifolds such that the heat kernels of manifolds in the family satisfy a uniform on-diagonal estimate from below, and we have established some basic results on (1) the structure of limit spaces of the family from the view-points of Gromov-Hausdorff distance and the spectral distance ; (2) the convergence of energy forms ; (3) the spectral convergence of vector bundle Laplacians ; (4) the convergence of harmonic maps of manifolds in the family to Riemannian manifolds of nonpositive curvature. The volume doubling condition and the Neumann-Poincare inequality, equivalently the parabolic Harnack inequality, play important roles in these investigations.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (40 results)

All Other

All Publications (40 results)

  • [Publications] A. Kasue-A. Mendori: "Riemarmian submersion and isoperimetric inegualifies"Geometriae Dedicata. 70. 27-47 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A. Kasue, H. Kumua-Y, Ogura: "Convergence of heat kernels on a compact Riema**** manifocd"Kyushu J. Math.. 51-3. 453-524 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Kato: "開リーアン多様体上にスカラー曲率の方程式"数学. 51-3. 225-240 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T, Hosimoto-K, Ohoa: "Cutingana Pastiug of Riemann surfaces with Avelian differential"International J. Math. (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. Nishio (et al): "Note on poly-supertemparatures on a stril domain"Osaka J. Math.. (予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A.Kasue and A.Mendori: "Riemannian submersions and isoperimetric constants"Geometriae Dedicata. 70. 27-47 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A.Kause, H.Kumura and Y.Ogura: "Convergence of heat kernels on a compact mainfold"Kyushu J. Math.. 51. 453-524 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Masuda, L.Moser-Jauslin and T.Petrie: "Invariants of equivariant algebraic vector bundles and inequalities for dominant weights"Topology. 37. 161-177 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Masuda, L.Moser-Jauslin and T.Petrie: "Equivariant algebraic vector bundles over cones with smooth one dimensional quotient"J.Math.Soc.of Japan. 50. 379-414 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Masuda and R.Schultz: "On the nonuniqueness of equivariant connected sums"J.Math.Soc.Japan. 51. 413-435 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kato, M.Umehara and K.Yamada: "An inverse problem of the flux for minimal surfaces"Indiana Univ. Math.J.. 46. 529-559 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kato: "Structure theorems of the scalar curvature equation on subdomains of a compact Riemannian manifold"Tsukuba J. Math.. 21. 153-168 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kato, M.Umehara and K.Yamada: "General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux"Comm.Anal.Geom. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kato: "Uniqueness of solutions of an elliptic singular boundary value problem"Osaka J.Math. 35. 279-302 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kato, M.Umehara and K.Yamada: "An inverse problem of the flux formula"Proceedings of the 4th Interanatinal Congress of Geometry (Thessaloniki, 1996), Giachoudis-Giapoulis, Thessaloniki. 228-233 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kato, M.Umehara and K.Yamada: "General existence of minimal surfaces with prescribed flux II"Topics in complex analysis, differential geometry and mathematical physics(St.Konstantin, 1996), World Sci.Publishing, River Edge, NJ. 116-135 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Kato: "Nonexistence of subsolutions of a nonlinear elliptic equation on bounded domains in a Riemannian manifold"Hiroshima Math.J.. 28. 419-435 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Hashimoto, Y.Yasui, S.Miyagi and T.Ootsuka: "Applications of the Ashtekar gravity to four-dimensional hyperk ahler geometry and Yang-Mills instantons"J.Math.Phys.. 38. 5833-5839 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Hashimoto: "On Macdonald's formula for the volume of a compact Lie group, Comment"Math.Helv.. 72. 1-3 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Hashimoto and K.Ohba: "Cutting and pasting of Riemann surfaces with Abelian differentials"International Journal of Mathematics. 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.Nishio, K.Shimomura and N.Suzuki: "Note on poly-supertemperatures on a strip domain"Osaka J. Math.. 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.Nishio, K.Shimomura and N.Suzuki: "A general form of mean value property of poly-temperatures on a strip domain"Proceedings of the 7th international colloquium on differential equations, D Bainov, ed. VSP, Utrecht. 269-276 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Nishio: "Capacity associated with an α-parabolic operator"Proceedings of the 7th international colloquium on differential equations, D Bainov, ed., VSP, Utrecht. 253-260 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Nishio: "Martin boundary at infinity for the heat equation"Proceedings of the 7th international colloquium on differential equations, D Bainov, ed., VSP, Utrecht. 261-267 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A.Kasue-A.Mendori: "Riemannian submersionsand isoperimetric inequalities"Geometriae Dedicata. 70. 27-47 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Masuda (et. al): "Equivariant algebraic vetor bundles over cones with smooth one dimensional quatient"J. Math. Soc. Japan. 50. 379-414 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] S. Kato: "開Riemann多様体上のスカラー曲率の方程式"数学. 51-3. 225-240 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y. Hashimoto-K.Ohba: "Cutting and pasting of Riemann surfaces with Abdimdiffential"International J. Math.. (予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Nishio (et al): "Note on poly-supertemperatures on a strip domain"Osaka J. Math.. (予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] A.Kasue-A.Mendori: "Riemannian submersions and isoperimetric inequalities" Geometriac Dedicata. 70. 27-47 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Masuda(et.al.): "Equivariant algebraic vector bundles over cones with smooth one-dimensional quotient" J.Math.Soc.Japan. 50・2. 379-414 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Kato: "Uniqueness of solutions of anelliptic singular bandary value problem" Osaka J.Math.35・2. 279-302 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Nishio(et al): "A mean value property of poly-temperatures on a strip domain" J.London Math.Soc.(予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Hashimoto-K.Ohba: "Cutting and pasting of Riemann surfoces with Abelian differentials" International J.Math.(予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Kasue, Y.Ogura H.Kumura: "Convergence of heatkevnels on a compact manifold" Kyushu J.Math.51・2. 453-524 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] A.Kasue, A.Mendori: "Riemannian submersions and iso permetric inefualities" Geometriae Dedicate. (to appear). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Kato, K.Yamada H.Umehara: "An inverse problem of the flux Forminimal surfaces" Indiana Univ.Math.J.46. 529-559 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Kato: "Unigueness of solutzons of an elliptic singular boundary value problem" Osaka J.Math.(to appear). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Nishio, N.Suzuki K.Shimomura: "A mean value property of pory-temperatures on a strip domein" J.London Nashi.Soc. (to appear). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Hashimoto(et al): "Applications of the Ashtekar graunty to four-dimensional" J.Math.Phys.38. 5833-5839 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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