2000 Fiscal Year Final Research Report Summary
Collapsing Theory of 4-manifolds and the geometry of Alexandrov Spaces
| Project/Area Number |
10440023
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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 | KYUSHU UNIVERSITY |
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Principal Investigator |
YAMAGUCHI Takao Kyushu-University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (00182444)
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| 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)
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| Project Period (FY) |
1998 – 2000
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| Keywords | collapsing / 4-manifolds / Gromov-Hausdorff distance / Alexandrov spaces / soul theorem |
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| 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.
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