1990 Fiscal Year Final Research Report Summary
Problems in Global Analysis
| Project/Area Number |
01460003
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Tohoku University |
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Principal Investigator |
KOTAKE Takeshi Tohoku Univ., Maths. Inst., Prof., 理学部, 教授 (30004427)
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| Co-Investigator(Kenkyū-buntansha) |
ARAI Hitoshi Tohoku Univ., Maths. Inst., Lecturer, 理学部, 講師 (10175953)
BANDO Shigetoshi Tohoku Univ., Maths. Inst., Assoc. Prof., 理学部, 助教授 (40165064)
ITO Hidekazu Tohoku Univ., Maths. Inst., Assoc. Prof., 理学部, 助教授 (90159905)
TAKAGI Izumi Tohoku Univ., Maths. Inst., Assoc. Prof., 理学部, 助教授 (40154744)
KATO Junji Tohoku Univ., Maths. Inst., Prof., 理学部, 教授 (80004290)
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| Project Period (FY) |
1989 – 1990
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| Keywords | Schrodinger operator / Dirac operator / integrable hamiltonian system / functional differential equation / harmonic function / Bergman kernel |
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| Research Abstract |
The research had the purpose of contributing to the cross-fertilisation of analysis and geometry, through the development of global analysis on manifolds, and exploring at the same time interesting and potentially fruitful interrelations between various fields of mathematics.
The project involved work in partial differential equations, dynamical systems, harmonic analysis and other diverse topics in analysis.
The results obtained during the period of this research project can be summarised as follows.
1. Partial differential equations and their applications to geometry : (1) study of asymptotic distribution of eigenvalues for Schrodinger operators with non-classical positive potentials ; (2) study of equivariant index of Dirac operators on spin manifolds ; (3) proof of the removability of isolated singularities for holomorphic vector bundles in connection with their Ricci curvature.
2. Dynamical system and functional differential equations : (1) reduction of integrable hamiltonian system to the normal form near singular points ; (2) discovery of various criteria for the existence and stability of functional differential equations with infinite delay.
3. Nonlinear analysis : (1) study on geometric structures of solutions, such as pattern formation and the appearance of singularities for reaction-diffusion equations ; (2) proof of asymptotic stability of gradient flow, associated with the Yang-Mills functional.
4. Harmonic analysis and operator theory : (1) proof of a Fatou-type theorem concerning the boundary behavior of harmonic functions on strictly pseudo-convex domains ; (2) study on the interrelation between the order structure and the regular completion of operator algebras.
5. Analysis on complex manifolds : study on the Bergman kernels on strictly pseudo-convex Reinhardt domains in C^2 in connection with the Chern-Moser invariant polynomials of the boundaries.
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