Deformations and Collapsing of Curved Spaces

该方案涉及研究具有不同曲率界的黎曼度量的变形和坍缩。一个感兴趣的基本对象是给定开流形上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，并且由诸如外科理论产生的非明显的微分同态相互拉紧。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制联系起来。为此目的，计划研究变形下灵魂的连续性，特别是对于具有余维二灵魂的几何相当刚性的流形。提出了一种获取低维流形模空间详细结构的方法。另一个课题是通过建立纤缩定理的相对版本来研究非负弯曲开放流形内灵魂的塌缩。另一个项目是分析有界负曲率的有限体积流形中不允许负缩紧度量的尖端截面的坍缩。从广义上讲，该方案试图理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。提出研究给定空间中所有可能几何形状的流形；人们对这种形状的流形的拓扑特征了解甚少。

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.