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Collapsing Theory of 4-manifolds and the geometry of Alexandrov Spaces

Research Project

Project/Area Number 10440023
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

YAMAGUCHI Takao  Kyushu-University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (00182444)

Co-Investigator(Kenkyū-buntansha) OOTSU Yukio  Osaka University, Graduate School of Science, assistant Prof., 大学院・理学研究科, 講師 (80233170)
SHIOHAMA Katsuhiyo  Sa Ga University, Department of Science and Technology, Prof., 理工学部, 教授 (20016059)
SATO Eiichi  Facalty of Mathematics, Prof., 大学院・数理学研究院, 教授 (10112278)
塩谷 隆  九州大学, 大学院・数理学研究科, 助教授 (90235507)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥6,500,000 (Direct Cost: ¥6,500,000)
Fiscal Year 2000: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 1999: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 1998: ¥2,300,000 (Direct Cost: ¥2,300,000)
Keywordscollapsing / 4-manifolds / Gromov-Hausdorff distance / Alexandrov spaces / soul theorem / 4次元多様体 / 曲率と位相 / 収束理論
Research Abstract

In this research from 1998 to 2000, we have decided the topology of collpsed Riemannian closed 4-manifolds with uniform lower bound on sectional curvature and upper diameter bound in terms of the singular fiber structure over the limit Alexandrov space. In the course of the proof of this result, we have also succeeded the classification of complete nonnegatively curved Alexandrov spaces of ditmensions three and four. The detail of the results are the following :
・We have constructed a local S^1-action on a collapsed 4-manifold when the limit space is of dimension three ;
・We have proved that a collapsed 4-manifold is either a sphere-bundle or a Seifert torus-bundle over the limit space ;
・In the case when the limit has dimension two and non-empty boundary, a decomposition of the collapsed 4-manifold along the boundary of the limit space has been obtained ;
・In the case when the limit is an interval, we have proved that the collapsed 4-manifold is a gluing of at least two and at most four disk-buundles ;
・We have proved a soul theorem for 4-dimensional concompact complete Alexandrov spaces with nonnegative curvature ;
・We have succeeded in proving a splitting theorem for complete Alexandrov spaces with nonnegative curvature and with disconnected boudary ;
・We have obtained a metric classification of compact Alexanadrov spaces with nonnegative curvature and with maximal vertices.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (32 results)

All Other

All Publications (32 results)

  • [Publications] Takao Yamaguchi: "Isometry groups of spaces with curvature bounded above"Math.Z. 232. 275-286 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takao Yamaguchi: "Collapsing Riemannian 4-manifolds."Sugaku. 52. 172-186 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takao Yamaguchi and Takashi Shioya: "Collapsing three-manifolds under a lower curvature bound"J.Differential Geometry. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takashi Shioya: "Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature"Manuscripta Math.. 99. 509-516 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takashi Shioya: "The limit spaces of two-dimensional manifolds with uniformly bounded inte-gral curvature"Trans.Amer.Math.Soc.. 351. 1765-1801 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takashi Shioya,: "Convergence of Alexandrov spaces and spectrum of Laplacian"J.Math.Soc.Japan. 53. 1-15 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Eiichi Sato and Yasuyuki Kachi: "Polarized varieties whose points are joined by rational curves of small degree"Illinois J.Math.. 43. 350-390 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Eiichi Sato and Zhao Yicai: "Smooth 4-folds which contains a P^1-bundle as an ample divisor"Manuscripta Math.. 101. 313-323 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Eiichi Sato: "Behavior of rational curves of the minimal degree in projective space bundle in any characteristic"J.Math.kyoto.univ.. 40. 675-706 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Katsuhiro Shiohama and Yoe Itokawa.: "The boundedness of certain minimal submanifolds of positively curved Rie-mannian spaces,"J Differential Geom.Appl.. 11. 97-104 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Katsuhiro Shiohama and Xu,Hong Wei: "Rigidity and sphere theorems for submanifolds II."Kyushu J.Math.. 54. 103-109 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Katsuhiro Siohama.: "Sphere Theorems,"Handbook of differential geometry.. 1. 865-903 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takao Yamaguchi: "Isometry groups of spaces with curvature bounded above"Math.Z.. 232. 275-286 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takao Yamaguchi: "Collapsing Riemannian 4-manifolds"Sugaku. 52. 172-186 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takao Yamaguchi and Takashi Shioya: "Collapsing three-manifolds under a lower curvature bound"J.Differential Geometry. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takashi Shioya: "Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature"Manuscripta Math.. 99. 509-516 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takashi Shioya: "The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature"Trans.Amer.Math.Soc.. 351. 1765-1801 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Takashi Shioya: "Convergence of Alexandrov spaces and spectrum of Laplacian"J.Math.Soc.Japan. 53. 1-15 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Eiichi Sato and Yasuyuki Kachi: "Polarized varieties whose points are joined by rational curves of small degree"Illinois J.Math.. 43. 350-390 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Eiichi Sato and Zhao Yicai: "Smooth 4-folds which contains a P^1-bundle as an ample divisor"Manuscripta Math.. 101. 313-323 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Eiichi Sato: "Behavior of rational curves of the minimal degree in projective space bundle in any characteristic"J.Math.kyoto.univ.. 40. 675-706 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Katsuhiro Shiohama and Yoe Itokawa: "The boundedness of certain minimal submanifolds of positively curved Riemannian spaces"J Differential Geom.. 11. 97-104 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Katsuhiro Shiohama and Xu, Hong Wei: "Rigidity and sphere theorems for submanifolds II"Kyushu J.Math.. 54. 103-109 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Katsuhiro Shiohama: "Sphere Theorems"Handbook of differential geometry. 1. 865-903 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Yamaguchi and T.Shioya: "Collapsing three-manifolds under a lower curvature bound, to appear"J.Differential Geometry..

    • Related Report
      2000 Annual Research Report
  • [Publications] 山口孝男: "4次元Riemann多様体の崩壊"数学. 52・2. 172-186 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.shiohama and X.H.Xu: "Rogidity and sphere theorems for submanifolds II"Kyushu J.Math.. 54. 103-109 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Shiohama: "Sphere Theorem"Handbook of Differential Geometry. 1. 865-903 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T, Yamaguchi.: "Isometry groups of spaces with curvature bounded above"Math. Z.. 232. 275-286 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T, Shioya.: "Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature."Manuscipta Math.. 99. 509-516 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K, Shiohama: "Sphere Theorem"Hand book of Differential Geometry.. Vol 1. 865-903 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 山口 孝男: "4次元Riemann多様体の崩壊"数学52巻. 第2号(発表予定). (2000)

    • Related Report
      1999 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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