A study of higher dimensional geometric variational problems

• Project/Area Number: 18540206
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: ISOBE Takeshi Tokyo Institute of Technology, 大学院・理工学研究科, 准教授 (10262255)
• Project Period (FY): 2006 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥2,450,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥450,000); Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2008: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2007: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2006: ¥500,000 (Direct Cost: ¥500,000)
• 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 / ヤンーミルズ方程式 / ヤンーミルズーディラック方程式 / 幾何学的変分問題 / コンパクト性 / トポロジー / Faddeev-Skyrme mode1 / 安定性 / ヤンーミルズ方程 / 臨界次元 / 正則性 / 変分問題 / ソボレフバンドル / Yang-Mills汎関数

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.

Research Products

• [Journal Article] Topological and analytical properties of Sobolev bundles (2009)
  • Author(s): Takeshi Isobe
  • Journal Title: The critical case. Ann. Global Anal. Geom. 35 Pages: 277-337
  • Peer Reviewed
• [Journal Article] Topological and analytical properties of Sobolev bundles, I : The critical case (2009)
  • Author(s): Takeshi Isobe
  • Journal Title: Ann.Globai Anal.Geom. 35 Pages: 277-337
  • Peer Reviewed
• [Journal Article] Topological and analytical properties of Sobolev bundles (2009)
  • Author(s): Takeshi Isobe
  • Journal Title: I : the critical case, Annals of Global Analysis and Geometry 35巻 Pages: 277-337
  • Peer Reviewed
• [Journal Article] A regularity result for a class of degenerate Yang-Mills connections in critical dimensions. (2008)
  • Author(s): Takeshi Isobe
  • Journal Title: Forum Math. 20 Pages: 1109-1139
  • Peer Reviewed
• [Journal Article] On a minimizing property of the Hopf soliton in the Faddeev-Skyrme model (2008)
  • Author(s): Takeshi Isobe
  • Journal Title: Rev. Math. Phy. 20 Pages: 765-786
  • Peer Reviewed
• [Journal Article] A regularity result for a class of degenerate Yang-Mills connections in critical dimensions (2008)
  • Author(s): Takeshi Isobe
  • Journal Title: Forum Math 20巻 Pages: 1109-1139
  • Peer Reviewed
• [Journal Article] On a minimizing property of the Hopf soliton in the Faddeev-Skyrme model (2008)
  • Author(s): Takeshi Isobe
  • Journal Title: Rev. Math. Phys. 20巻 Pages: 765-786
  • Peer Reviewed
• [Journal Article] On a minimizing property of the Hopf soliton in the Faddeev・Skyrme model (2008)
  • Author(s): Takeshi Isobe
  • Journal Title: Reviews in Mathematical Physics 20 Pages: 765-786
  • Peer Reviewed
• [Journal Article] A regularity result for a class of degenerate Yang-Mills connections in critical dimensions (2008)
  • Author(s): Takeshi Isobe
  • Journal Title: Forum Mathematicum 20 Pages: 1109-1139
  • Peer Reviewed
• [Journal Article] Topology of Sobolev bundles (2007)
  • Author(s): Takeshi Isobe
  • Journal Title: 数理解析研究所講究録 1528 Pages: 104-116
• [Journal Article] Topology of Sobolev bundles (2007)
  • Author(s): Takeshi Isobe
  • Journal Title: 数理解析研究所講究録 1528巻 Pages: 104-116
  • Peer Reviewed
• [Journal Article] Topology of Sobolev bundles (2007)
  • Author(s): 磯部 健志
  • Journal Title: 京都大学数理解析研究所講究録 No.1528 Pages: 104-116
• [Journal Article] Topology of Sobolev bundles (2007)
  • Author(s): 磯部 健志
  • Journal Title: 数理解析研究所講究録 1528 Pages: 104-116
• [Journal Article] On global singularities of Sobolev mappings (2006)
  • Author(s): Takeshi Isobe
  • Journal Title: Math. Z. 252 Pages: 691-730
  • Peer Reviewed
• [Journal Article] On global singularities of Sobolev mappings (2006)
  • Author(s): Takeshi Isobe
  • Journal Title: Math. Z. 252巻 Pages: 691-730
  • Peer Reviewed
• [Journal Article] On global singularities of Sobolev Mappings (2006)
  • Author(s): Takashi Isobe
  • Journal Title: Mathematische Zeitschrift 252 Pages: 691-730
• [Presentation] Topology of Sobolev bundles (2006)
  • Author(s): 磯部健志
  • Organizer: 変分問題とその周辺
  • Place of Presentation: 京都大学数理解析研究所