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
| Project/Area Number |
13640169
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Osaka University |
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Principal Investigator |
TAKEGOSHI Kensho Osaka University Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (20188171)
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| 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)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Parabolicity of manifold / Harmonic map / The scaler curvature equation / Subharmonic functions |
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| 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).
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