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Differential Geometric Reserch on Manifolds

Research Project

Project/Area Number 07304006
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section総合
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

KENMOTSU Katsuei  Tohoku Univ.Math.Inst.Prof., 大学院・理学研究科, 教授 (60004404)

Co-Investigator(Kenkyū-buntansha) YAMADA Kotaro  Kumamoto Univ.Dep.of Math.Asso.Prof., 教養部, 助教授 (10221657)
KASUE Atsushi  Osaka City Univ.Dep.of Math.Prof., 理学部, 教授 (40152657)
FUKAYA Kenji  Kyoto Univ.Math.Inst.Prof., 理学研究科, 教授 (30165261)
MIYAOKA Reiko  Tokyo Inst.Tech.Dep.of Math.Prof., 理学部, 助教授 (70108182)
OGIUE Koichi  Tokyo Metro.Univ.Dep.of Math.Prof., 理学部, 教授 (10087025)
酒井 隆  岡山大学, 理学部, 教授 (70005809)
西川 青季  東北大学, 大学院・理学研究科, 教授 (60004488)
Project Period (FY) 1995 – 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥8,300,000 (Direct Cost: ¥8,300,000)
Fiscal Year 1996: ¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 1995: ¥4,600,000 (Direct Cost: ¥4,600,000)
Keywordsmanifold / Differential Geometry / リーマン幾何学 / シンプレクテイック幾何学 / 極小曲面 / 等質空間 / 幾何学的変分問題 / Floerホモロジー
Research Abstract

Kenmotsu has published a paper, in which he proved theorems for intersections of minimal submanifolds in manifolds with partially positive curvature. Kenmotsu is studying local behavior of the Kaehler angles of minimal surfaces with constant Gaussian curvature in two dimensional complex space forms : In order to classify such minimal surfaces, at first we obtained differential geometric characterization of the second fundamental forms of such minimal surfaces. By using it we obtained an overdetermined system for the Kaehler angle. This is reduced to a system of two ODE's. By the values of the Gaussian curvature of the surface and the curvature of ambiant space, these systems are different. We developed analysis to these systems extensively and proved that they have no non trivial common solution even locally. It implies local classification theorem of suchminimal surfaces.
Fukaya has proved the Arnold conjecture in the general setting. This is really exciting.
For reserch of submanifold geometry, R.Miyaoka has studied relations between minimal surfaces in complex projective spaces and the Toda equations extensively and published her results in the Crelle Journal. Yamada has contributed to construct the theory of constant mean curvature surfaces in the hyperbolic spaces.
For the reserch of global Riemannan geometry, Suyama has given a new method to construct diffeotopy of standard spheres and applying it he proved a differentiable pinching theorem for 0.654 pinched compact riemannan manioflds. T.Sakai has written a textbook of global Riemannian geometry which was published by the American Math.Society.

Report

(3 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • 1995 Annual Research Report
  • Research Products

    (33 results)

All Other

All Publications (33 results)

  • [Publications] K.Kenmotsu: "Hadamard-Frankeltype theoums for maniyelds with patislly poietine carvatus" Pacific Journal of Mathumatics 22GD01:176. 129-139 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] R.Miyaoka: "Flefamety of isonetic superconfermal harmony maps and affine Tocla equations" Teur.for die reine und angewante Math.481. 1-25 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] K.Fukaya: "Floer homalogy of connacted sum of homology 3-splues" Topolagy. 35. 88-136 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] K.Fukaya: "Merse homotopy and Chern-Sinons Pertucbation theary" Comm.of Math.Phys. 81. 37-90 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] A.Kasue: "Spectral consequence of Riemannian mainfolds II" Tohpfsu Math.Journ.48. 71-120 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] M.Umehara: "Surfaces of constant mean cururtue C in H^3 (-C^2) anih prescrihecl hyp." Mathonatiachl Annalen. 304. 203-224 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 深谷 賢治: "解析力学と微分形式" 岩波書店, 160 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 酒井 隆: "Riemannian Geometry" Amer.Math.Soc., 356 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kenji Fukaya: "Morse homotopy and Chern-Simons Perturbation theory" Commun.of Math.Phys.81. 37-90 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Atsushi Kasue: "Convergence of Riemannian manifolds and Albanese tori" Osaka J.Math.32. 677-688 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Atsushi Kasue: "Spectral convergence of Riemannian manifolds II.(With H.Kumura)" Tohoku Math.J.48. 71-120 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Katsuei Kenmotsu: "Intersections of Minimal Submanifolds in Manifolds with Partially Positive Curvature (With C.Y.Xia)" Kodai Math.Jour.18. 242-249 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Katsuei Kenmotsu: "Hadamard-Frankel type theorems for minimal submanifolds (With C.Y.Xia)" Pacific Journ.Math.176. 129-139 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Katsuei Kenmotsu: "Some recent topics in surfaces theory" Proc.Top.and Geo.Reserch Center, edited by II Bong Jung and Sang Hun Lee, TGRC-KOSEF,Taegue. 147-184 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Reiko Miyaoka: "The family of isometric superconformal harmonic maps and the affine Toda equations" Journal fur die reine und angewante Mathematik. 48. 1-25 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Koichi Ogiue: "Extrinsic shape of circles and the standard imbeddings of a Cayley projective plane (With T.Adachi, S.O'Maeda)" Hokkaido Math.J.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kotaro Yamada: "Surfaces of constant mean curvature c in H with prescribed hyperbolic Gauss map. (With M.Umehara)" Math.Annalen. 304. 203-224 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kotaro Yamada: "A duality on CMC-1 surface in the hyperbolic 3-space and a hyperbolic analogue of the Osserman Inequality (With M.Umehara)" Tsukuba Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Kenji Fukaya: Symplectic S-cobordism conjecture-summary-, in "Geometry and Physics" Lecture notes in pure and applied mathematics No.184 ed.by J.Andersen, J.Dupont, H.Pertersen, A.Swan. Marcel-Dekker, New-York, 209-220 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] K. Kenmotsu: "Hadamard-Frankel type theorems for manifolels with partially pasitinc curuntail" Pacific Journal of Mathematics. 176. 129-139 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] R. Miyaoka: "The family of isometric super conformal harmanic maps and affine Toda egoations" Jour. fur die reine und avgewante Math.481. 1-25 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] K. Fukaya: "Floer homalogy of connected sum of homelogy 3-sphees" Topology. 35. 88-136 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] K. Fukaya: "Morse homotopy and Chern-Simous Perturbation theory" Cominun. of Math. Phys.81. 37-90 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] A. Kasue: "Spectral conuergence of Riemannian manipoldsII" Tohoku Math. Journ.48. 71-120 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] M. Umehara: "Surfaces of constant mean curnative c in H^3 (-c) with presribed hyp. homes map" Mathamatische Annalen. 304. 203-224 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] 深谷賢治: "解析力学と微分型式" 岩波書店, 160 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] 酒井隆: "Riemannian Geomety" Awer. Math. Soc., 356 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] K.Fukaya: "Floer homology of connected sum of homology 3-spheres" Topology. 35. 88-136 (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] T.Sakai: "On Riemannian manfields admitting a function whose gradiont is of comt norm." Kodai Math,J. 19(印刷中). (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] A.Kasue: "Convergence of Riemannian manifelds and Albanese tori" Osaka J.Math.,. 32. 677-683 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] M.Umehara and K.Yamada: "Surfaces of constant mean curuature in H^3(-C^2)with psesuibed hyp Gauss map" Mathematische Annalen. (発表予定).

    • Related Report
      1995 Annual Research Report
  • [Publications] K Kenmotsu and C.Xia: "Intersections of minimal submanifelds in Manifolds of Partially Pesilie Curvature" Kodai Math.J.,. 18. 242-249 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] S.Nishikawa and R.Schoen: "Lectures on Geometric Variational Problems" Springer Verlag, 152 (1996)

    • Related Report
      1995 Annual Research Report

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Published: 1995-04-01   Modified: 2016-04-21  

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