1998 Fiscal Year Final Research Report Summary
Various problems in Hamiltonian dynamical systems and related topics in geometry and analysis
| Project/Area Number |
09440050
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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 |
解析学
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| Research Institution | TOKYO INSTITUTE OF TECHNOLOGY |
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Principal Investigator |
ITO Hidekazu Associate Professor at Tokyo Institute of Technology Guraduate School of Science and Engineering, 大学院・理工学研究科, 助教授 (90159905)
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| Co-Investigator(Kenkyū-buntansha) |
ONO Kaoru Professor at Hokkaido University Guraduate School of Science, 大学院・理学研究科, 教授 (20204232)
TANAKA Kazunaga Associate Professor at Waseda University School of Science and Engineering,, 理工学部, 助教授 (20188288)
MORITA Takehiko Associate Professor at Tokyo Institute of Technology Guraduate School of Science, 大学院・理工学研究科, 助教授 (00192782)
MIYAOKA Reiko Associate Professor at Tokyo Institute of Technology Guraduate School of Science, 大学院・理工学研究科, 助教授 (70108182)
SHIGA Hiroshige Associate Professor at Tokyo Institute of Technology Guraduate School of Science, 大学院・理工学研究科, 助教授 (10154189)
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| Project Period (FY) |
1997 – 1998
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| Keywords | hamiltonian system / integrable system / variational method / symplectic geometry / zeta funetion / complex dynamical system |
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| Research Abstract |
The following is the abstract for the main results obtained under this research project.
1. In the research of integrable systems, Miyaoka proved that all isoparametric hypersurfaces in the sphere with six principal curvatures are homogeneous. For the proof, she used the isospectrality of the family of the shape operators on the focal set of isoparametric hypersurfaces. This is a remarkable result to solve the conjecture by Yau, and shows close connection between the theory of hypersurfaces and that of integrable systems.
2. In the research of ergodic theory, using transfer operators method, Morita obtained Fredholm determinant representation for the Selberg zeta function Z(s) of closed geodesics on hyperbolic Riemann surface with finite area. He investigated the spectral properties of the transfer operators, and then obtained some analytic information of Z(s).
3. By using variational methods, Tanaka studied unbounded solutions of singular Hamiltonian systems of the two-body type. Namely he proved the existence of a hyperbolic-like solutions for a class of singular potentials with the strong force alpha > 2. Also, he studied the prescribed energy problem for Hamiltonian systems of the same form with alpha = 2, and showed variationally the existence of periodic solutions on the zero energy surface. This led to an existence theorem of closed geodesics on noncompact Riemannian manifold.
4. In the research of symplectic geometry, Ono solved the Arnold's conjecture for general closed symplectic manifold by constructing Floer homology for periodic Hamiltonian and Gromov-Witten invariant. He generalized this approach further and studied constructions of Floer homology for Lagrangian intersection.
5. In the research of complex dynamical systems, Shiga showed some similar properties between limit sets of Kleinian group and Julia sets in theory of complex dynamical systems.
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