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

4流形的塌缩理论和Alexandrov空间的几何

• 批准号: 10440023
• 负责人: YAMAGUCHI Takao
• 金额: $ 4.16万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440023/
• 关键词: collapsing; 4-manifolds; Gromov-Hausdorff distance; Alexandrov spaces; soul theorem; 4次元多様体; 曲率と位相; 収束理論

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.

在1998年至2000年的研究中，我们确定了极限Alexandrov空间上奇异纤维结构的具有截面曲率一致下界和直径一致上界的坍缩riemann闭4流形的拓扑结构。在证明这一结果的过程中，我们还成功地分类了3维和4维的完全非负弯曲Alexandrov空间。具体结果如下：·当极限空间为三维时，我们构造了坍缩4流形上的局部S^1作用；我们证明了一个坍缩的4流形在极限空间上要么是球束，要么是塞弗特环束；在极限为二维且非空边界的情况下，得到了沿极限空间边界的折叠4流形的分解；在极限为区间的情况下，证明了坍缩的4流形是至少2个且最多4个盘束的胶合；我们证明了具有非负曲率的四维紧致完全亚历山德罗夫空间的一个灵魂定理；·我们成功地证明了具有非负曲率和不连通边界的完全亚历山德罗夫空间的分裂定理；我们获得了非负曲率且顶点极大的紧化Alexanadrov空间的度量分类。

项目成果

山口孝男: "4次元Riemann多様体の崩壊"数学. 52・2. 172-186 (2000)

Takao Yamaguchi：“4 维黎曼流形的塌陷” 数学 52・2。

Takao Yamaguchi and Takashi Shioya: "Collapsing three-manifolds under a lower curvature bound"J.Differential Geometry. (to appear).

Takao Yamaguchi 和 Takashi Shioya：“在曲率下限下折叠三流形”J.微分几何。

Katsuhiro Shiohama: "Sphere Theorems"Handbook of differential geometry. 1. 865-903 (2000)

Katsuhiro Shiohama：《球面定理》微分几何手册。

Takao Yamaguchi: "Collapsing Riemannian 4-manifolds."Sugaku. 52. 172-186 (2000)

Takao Yamaguchi：“折叠黎曼四流形。”Sugaku。

T.Yamaguchi and T.Shioya: "Collapsing three-manifolds under a lower curvature bound, to appear"J.Differential Geometry..

T.Yamaguchi 和 T.Shioya：“在曲率下限下折叠三流形，出现”J.Differential Geometry..