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2002 Fiscal Year Final Research Report Summary

Asymptotic behavior of solutions of a certain quasi non-linear operator and its application to geometric function theory

Research Project

Project/Area Number 13640169
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOsaka University

Principal Investigator

TAKEGOSHI Kensho  Osaka University Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (20188171)

Co-Investigator(Kenkyū-buntansha) KOISO Norihito  Osaka University Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70116028)
MABUCHI Toshiki  Osaka University Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80116102)
NAMBA Makoto  Osaka University Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004462)
SUGIMOTO Mitsuru  Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (60196756)
ENOKI Ichiro  Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (20146806)
Project Period (FY) 2001 – 2002
KeywordsParabolicity of manifold / Harmonic map / The scaler curvature equation / Subharmonic functions
Research Abstract

The purpose of this project is to study asymptotic behaviour of (sub-) solutions of a certain quasi non-linear operator P on a complete Riemannian manifold (M, g). Here P is either the Laplacian or the mean curvature operator which is the most interesting case. Several topics related to maximum principle for solutions of that operator have been studied. We could show the generalized maximum principle for such an operator P without any Ricci curvature condition of (M, g). Our method depends only on some volume growth condition of that manifold. From the principle we can induce several interesting results related to (1) uniqueness of solutions of the scaler curvature equation, (2) Liouville type theorem for harmonic maps, (3) isometric property of conformal transformations preserving scaler curvature and (4) value distribution of minimal immersions of complete manifolds, which contain almost all known results up to now in Riemannian geometry. Furthermore we studied a growth property of L^p-integrals of subharmonic functions on geodesic spheres on (M, g), and obtained an optimal growth estimate of those integrals. This result is also related to the maximum principle on complete manifolds. From this estimate we can yield a very simple and function theoretic proof for (M, g) to be parabolic, and get several results related to the problem (1)〜(4).

  • Research Products

    (9 results)

All Other

All Publications (9 results)

  • [Publications] Takegoshi, K.: "Strongly p-subharmonic functions and volume growth property of complete Riemannian manifolds"Osaka J Math.. 38. 839-850 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mabuchi, T.: "Heat kernel estimates and the Green functions on multiplier Hermitian manifolds"Tohoku Math. J.. 54. 261-277 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mabuchi, T: "A theorem of Calabi-Matsusima's type"Osaka J Math.. 39. 49-57 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Koiso, N.: "Convergence towards an elastica in a Riemannian manifold"Osaka J Math.. 37. 467-487 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kensho Takegoshi: "Torsion freeness theorems for higher direct images of canonical sheaves by a certain convex Keahler morphism"Osaka J. Math.. 36. 17-26 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kensho Takegoshi: "Strongly p-subharmonic functions and volume growth property of complete Riemannian manifolds"Osaka J. Math.. 38. 839-850 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Toshiki Mabuchi: "Heat kernel estimates and the Green functions multipler Hermitian manifolds"Tohoku Math. J.. 54. 261-277 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Toshiki Mabuchi: "A theorem of calabi-Matsusima's type"Osaka. J. Math.. 39. 49-57 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Norihito Koiso: "Convergence towards an elastica in a Riemannian manifold"Osaka J. Math.. 37. 467-487 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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