2009 Fiscal Year Final Research Report
A study of higher dimensional geometric variational problems
| Project/Area Number |
18540206
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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 | Tokyo Institute of Technology |
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Principal Investigator |
ISOBE Takeshi Tokyo Institute of Technology, 大学院・理工学研究科, 准教授 (10262255)
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| Project Period (FY) |
2006 – 2009
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| Keywords | ソボレフ空間 / 特異点 / 調和写像 / ヤンーミルズ汎関数 / Faddeev-Skyrme模型 / モース理論 / Yang-Mills-Dirac方程式 / エネルギー量子化 |
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| Research Abstract |
The purpose of this research is to give a functional analytic foundation for higher dimensional geometric variational problems defined on topologically non-trivial function spaces. The main results we have obtained are 1) We have classified singularities of Sobolev class mappings defined between manifolds. In particular, we have given a general theory of global singularities of Sobolev mappings. 2) We studied interrelations between topological and analytical properties of Sobolev bundles. We applied these results to higher dimensional gauge theory. 3) We proved the strong stability of Hopf soliton for the Faddeev-Skyrme model on the 3-sphere. 4) We proved a regularity of weak solutions for a certain kind of degenerate Yang-Mills connections in critical dimensions. 5) We established a Morse theory for the Yang-Mills-Dirichlet problem on 4-manifolds with boundary. 6) We proved a regularity and energy quantization for the Yang-Mills-Dirac equations defined on 4-manifolds.
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