Deformations and Collapsing of Curved Spaces

弯曲空间的变形和塌陷

• 批准号: 1105045
• 负责人: Igor Belegradek
• 金额: $ 14.19万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105045&HistoricalAwards=false
• 关键词: Deformations; Collapsing; Curved; Spaces

This proposal involves studying deformations and collapsing of Riemannian metrics with various curvature bounds. A basic object of interest is the moduli space of complete nonnegatively curved metrics on a given open manifold. Nearby points in the moduli space often correspond to metrics that may appear unrealated and are pulled next to each other by non-obvious diffeomorphisms produced e.g. by surgery theory. Thus to detect nontrivial topology in the moduli space one needs to relate comparison geometry input with machinery of algebraic topology. To this end it is planned to investigate continuity of souls under deformation, especially for manifolds with codimension two souls where the geometry is rather rigid. It is proposed to obtain detailed structure of the moduli space for low-dimensional manifolds. Another topic is studying collapse of souls inside nonnegatively curved open manifolds by establishing relative versions of the fibration theorems. Yet another project is to analyze collapse of cusp cross-sections in finite volume manifolds of bounded negative curvature that admit no negatively pinched metrics.In broad terms the proposal seeks to understand how curvature of the space controls its deformations and degenerations. The spaces considered range from boundaries of three-dimensional convex bodies to their higher dimensional analogs. It is proposed to investigate the manifold of all possible geometric shapes that a given space can assume; the topological features of this manifold of shapes are poorly understood.

这个建议涉及研究变形和倒塌的黎曼度量与各种曲率的界限。一个基本的感兴趣的对象是一个给定的开流形上的完全非负弯曲度量的模空间。模空间中的邻近点通常对应于可能看起来不相关的度量，并且通过例如由手术理论产生的非明显的同构而彼此相邻。因此，为了检测模空间中的非平凡拓扑，需要将比较几何输入与代数拓扑机器相关联。为此，计划调查连续性的灵魂变形下，特别是为流形余维两个灵魂的几何形状是相当刚性。提出了一种求低维流形模空间的详细结构的方法。另一个主题是通过建立纤维化定理的相关版本来研究非负弯曲开流形中灵魂的坍缩。另一个项目是分析有限体积流形中尖点截面的坍缩，该流形具有有界负曲率，不允许负收缩度量。广义上说，该提案旨在了解空间的曲率如何控制其变形和退化。所考虑的空间范围从三维凸体的边界到它们的高维类似物。有人建议研究给定空间可以假设的所有可能的几何形状的流形;这种形状流形的拓扑特征知之甚少。