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Various problems in Hamiltonian dynamical systems and related topics in geometry and analysis

Research Project

Project/Area Number 09440050
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionTOKYO INSTITUTE OF TECHNOLOGY

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)

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)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥4,200,000 (Direct Cost: ¥4,200,000)
Fiscal Year 1998: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1997: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywordshamiltonian system / integrable system / variational method / symplectic geometry / zeta funetion / complex dynamical system / シンプレティック幾何学 / エルゴード埋論 / 大域幾何学
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.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (39 results)

All Other

All Publications (39 results)

  • [Publications] Ito, Hidekazu: "Integrability of symplectic maps and their Birkhoff normal forms" Tohoku Math. J.49. 73-114 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shiga, Hiroshige: "On the monodromies of holomorphic families of Riemann surfaces and modular transformations" Math. Proc. Cambridge Philos. Soc.122. 541-549 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Miyaoka, Reiko: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces" Tohoku Math. J.51. 35-53 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Morita, Takehiko: "Markov systems and transfer operators asseciated with cofinite Fuchsion groups" Evgod. Th. & Dynam. Sys.17. 1147-1181 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Tanaka, Kazunaga: "Multiple positive solntions for some nonlinear elliptic systems" Topological Methods in Nonlinear Analysis. 10. 15-45 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Ono, Kaoru: "On Arnold's conjecture for symplectic fired points" Banach Center Publications. 45. 13-24 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Connolly, F, Van, Le Hong, and Ono, K: "Almost Complex Structures which Are Compatible with Kahler or Symplectic Structures" Ann. Global Analysis and Geometry. 15. 325-334 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 伊藤, 秀一: "常微分方程式と解析力学" 共立出版, 292 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 志賀, 啓成: "複素解析学 I=基礎理論" 培風館, 152 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Ito: "Integrable symplectic maps and their Birkhoff normal forms" Tohoku Math.J.49. 73-114 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Shiga: "On the mondromies of folomorphic families of Riemann surfaces and modular transformations" Math.Proc.Cambrigde Philos.Soc.122. 541-549 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Shiga and H.Tanigawa: "Projective structures on Riemann surfaces with discontinuous holonomies" Trans.Amer.Math.Soc.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Shiga: "On complex analytic properties of limit sets and Julia sets" preprint, Titech-Math 10-98 (#86). (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] R.Miyaoka: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces" Tohoku Math.J. 51. 35-53 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] R.Miyaoka: "The homogeneity of isoparametric hypersurfaces with six principal curvatures" Preprint, Titech-Math 04-98(#80). (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] I.Morita: "Markov systems and transfer operators associated with cofinite Fuchsian groups" Ergod.Th.& Dynam.Sys.17(5). 1147-1181 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Nomura: "Absence of diffusion near the bottom of the spectrum for a random Schrodinger operator on L^2 (R^3)" J.Math.Kyoto Univ.37(4). 639-687 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Ambrosetti and K.Tanaka: "On Keplerian N-body type problems, in "Nonlinear Analysis and continuum mechanics" Paper for the 65-th birthday of James Serrin" (G.Buttazzo, G.P.Galdi, E.Lanconelli, P.Pucci ed.). Springer. 15-25 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Tanaka: "Multiple positive solutions for some nonlinear elliptic systems" Topological Methods in Nonlinear Analysis. 10. 15-45 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] P.Felmer and K.Tanaka: "Hyperbolic-like solutions for singular Hamiltonian systems" NoDEA Nonlinear Differential Equations Appl.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Tanaka: "Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds" Ann.Inst.H.Poincare Anal.Non Lineaire. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Kabeya and K.Tanaka: "Uniqueness of positive radial solutions of semilinear elliptic equations in R^N and Sere's non-degeneracy condition" Comm.Partial Differential Equations. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ono, F.Connolly and H.-V.Le: "Almost complex structure which are compatible with Kahler or symplectic structures" Ann.Global Anal.Geometry. 15. 325-334 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ono: "On Arnold's conjecture for symplectic fixed points" Banach Center Publications 45"Homotopy and Geometry"(J.Oprea and A.Tralle eds.). 13-24 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Fukaya and K.Ono: "The Arnold conjecture and Gromov-Witten invariant" Topology. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shiga,Hiroshige: "Projective structures on Riemann surfaces with discontinuous holonomies" Trans.Amer.Math.Soc. (出版予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] Miyaoka,Reiko: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces" Tohoku Math.J.51. 35-53 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Felmer,Patricio: "Hyperbolic-lihe solutions for singular hamiltonian systems" NoDEA Nonlinear Differential Equations Appl.(出版予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] Tanaka,Kazunaga: "Periodic solutions for singulor Hamiltonian systems and closed geodesics on non-compact Riemannion manifolds" Ann.Inst.H.Poincare(Non linear). (出版予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] Ono,Kaoru: "On Arnold's conjecture for symplectic fixed points" Banach Center Publications. 45. 13-24 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Fukaya,Kenji: "The Arnold conjectnre and Gromov-Witten in variant" Topology. (出版予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] Shiga,Hiroshige: "On the monodromies of holomorphic families of Riemonn surfaces and modular transformations" Math.Proc.Cambridge Philes.Soc.122(3). 541-549 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Miyaoka,Reiko: "The splitting and deformations of the generalized Gauss map of compact CMC surfaces" Tohoku Math.J.(出版予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Morita,Takehiko: "Markov systems and transfor operaters associated with Fuchsian groups" Ergod.Th.& Dynam.Sys.17. 1147-1181 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Tanaka,Kazunaga: "Multiple positive solutions for some nonlinear elliptic systems" Topological Methods in Nonlinear Analysis. 10(出版予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Tanaka,Kazunaga: "On Keplerian N-body type problems" "Nonlinear Analysis and Continuum mechanics"(springor)ed : G.Buttazzo,G.P,Galdi,E.Lanconelli,P.Pucci. 15-25 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Ono,Kaoru: "On Arnold's conjecture for symplectic fixed points" Banach Center Publications. (出版予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] 伊藤,秀一: "常微分方程式と解析力学" 共立出版, 292頁 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 志賀,啓成: "複素解析学I=基礎理論" 培風館, 152頁 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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