1996 Fiscal Year Final Research Report Summary
Study of Nonlinear problems in Geometry
| Project/Area Number |
07454012
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tohoku University |
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Principal Investigator |
NISHIKAWA Seiki Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004488)
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| Co-Investigator(Kenkyū-buntansha) |
ARAI Hitoshi Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10175953)
HORIHATA Kazuhiro Tohoku University, Graduate School of Science, Research Associate, 大学院・理学研究科, 助手 (10229239)
NAYATAMI Shin Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70222180)
NAKAGAWA Yasuhiro Tohoku University, Graduate School of Science, Research Associate, 大学院・理学研究科, 助手 (90250662)
BANDO Shigetoshi Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40165064)
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| Project Period (FY) |
1995 – 1996
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| Keywords | Geometry of manifolds / Nonlinear problem / Geometric variational problem / Harmonic map / Moduli space / Einstein-Kaehler metric / Strongly pseudo-convex CR manifold / 強擬凸CR多様体 |
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| Research Abstract |
The purpose of this project is to study Nonlinear Problems, arising in researches of deformation of various geometric structures and their moduli spaces, from the point of view of Geometric Variational Problems. The following is the summary of principal results obtained under the project.
1. Nishikawa studied the Dirichlet problem at infinity for harmonic maps between general k-term Carnot spaces, which are homogeneous spaces of negative curvature obtained as solvable extension of k-step nilpotent Carnot Lie groups. He found the necessary conditions for the boundary values on ideal boundaries, and proved the existence and uniqueness of solutions when given boundary values are nondegenerate.
2. Bando studied the degeneration phenomena od Hermitian-Einstein metrics on stable holomorphic vector bundles over a compact Kaehler manifold, and proved that the moduli spaces of these bundles can be compactified by adding reflexive sheaves as their boundaries.
3. Nakagawa studied the existence problem of Einstein-Kaehler metrics, and proved combrinatiorial formulae describing the Futaki invariants and generalized Killing forms on toric Fano orbifolds terms of data read off from their corresponding convex bodies.
4. Nayatani constructed in a standard way pseudo-Riemannian metrics compatible with the pseudo-conformal structures on the ideal boundaries of rank one locally Riemannian symmetric space of noncompact type.
5. Horihata studied the nonlinear parabolic system of partial differential equations associated with harmonic map, and proved the partial regularity of weak solutions based on the monotonicity formula in the case when the space is 3 dimensional.
6. Arai proved the best possible estimate for solutions of pseudo-differential equations on nilpotent Lie groups, and applied it to obtain the best possible estimate for solutions of the tangential Cauchy-Riemann equation on strongly pseudo-convex CR manifolds.
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