1999 Fiscal Year Final Research Report Summary
n-harmonic maps and conformal structures on manifolds
| Project/Area Number |
10640209
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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 |
Global analysis
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| Research Institution | Yamaguchi University |
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Principal Investigator |
NKAUCHI Nobumitsu Yamaguchi University, Faculty of Science, associate professor, 理学部, 助教授 (50180237)
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| Co-Investigator(Kenkyū-buntansha) |
TAKAKUWA Shoichiro Tokyo Metropolitan University, Faculty of Science, associate professor, 理学部, 助教授 (10183435)
TAKEUCHI Hiroshi Shikoku University, Faculty of Management and Information Science, professor, 経営情報学部, 教授 (20197271)
KAWAI Shigeo Saga University, Faculty of Culture and Education, professor, 文化教育学部, 教授 (30186043)
KATO Shin Osaka City University, Faculty of Science, associate professor, 理学部, 助教授 (10243354)
KOBAYASHI Osamu Kanazawa University, Faculty of Science, professor, 理学部, 教授 (10153595)
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| Project Period (FY) |
1998 – 1999
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| Keywords | n-harmonic map / p-harmonic map / conformal structure / manifold / variational problem |
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| Research Abstract |
With applications to the geometry of conformal structures in mind, we studied p-harmonic maps and n-harmonic maps. From some points of view, we considered the case in which the source or the target is the standard sphere. We obtained a regularity result for p-harmonic maps into the sphere. Though the concept of p-harmonic maps are, speaking formally, a generalization of that of harmonic maps, we are often in the face of different features and some difficulties. We give the following summary of what are main difficulties in the study of p-harmonic maps :
(1) The p-harmonic map equation is ellptic, but degenerate unless p = 2.
(2) In several cases, there exist some nontrivial terms which vanishes only when p = 2
(3) The exponent or the index "2" in the case of harmonic maps (p = 2) has various different meanings for general p, for example, p, 2p - 2, …, which are all equal to 2 when p = 2.
Some cases in the above difficulties can be controled with some tricks. In the Bochner-Weitzenbock formula, for example, we can make nontrival terms in case of p ≠ 2 vanish in a certain integral formula. As for the rest cases in the above difficulties, new techniques are necessary and we have a hope of them in the study in the future.
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