Project/Area Number  10640209 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Global analysis

Research Institution  Yamaguchi University 
Principal Investigator 
NKAUCHI Nobumitsu Yamaguchi University, Faculty of Science, associate professor, 理学部, 助教授 (50180237)

CoInvestigator(Kenkyūbuntansha) 
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)
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)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,400,000 (Direct Cost : ¥3,400,000)
Fiscal Year 1999 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1998 : ¥2,100,000 (Direct Cost : ¥2,100,000)

Keywords  nharmonic map / pharmonic map / conformal structure / manifold / variational problem / n調和写像 / p調和写像 / 共形幾何構造 / 多様体 / 変分問題 
Research Abstract 
With applications to the geometry of conformal structures in mind, we studied pharmonic maps and nharmonic 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 pharmonic maps into the sphere. Though the concept of pharmonic 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 pharmonic maps : (1) The pharmonic 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 BochnerWeitzenbock 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.
