Collapsing, pinching, and curvature bounds

项目编号：dms -0503864项目负责人：Igor belegradek本课题研究弯曲黎曼流形的拓扑结构。一个目标是研究坍缩流形，即在一定小但确定的尺度上呈现较低维度的流形。具体而言，计划研究几乎非负弯曲流形，有界非负曲率的坍缩流形以及有限体积非正弯曲流形的尖端截面。在一个稍微不同的方向上，提出了研究负压缩流形，包括压缩估计，最优压缩情况下的刚性，压缩假设的有限性定理，星象球的渐近几何和负压缩群。该项目继续了Belegradek和合作者的早期工作，特别是最近与Kapovitch关于具有可适应基群的负缩形的微分同态分类（解决了bowditch的问题），以及对这种流形的最优缩形估计（回答了Gromov的问题）的结果。虽然该项目主要涉及整体黎曼几何的内在发展，但它也在一定程度上受到科学和工程的推动，其中弯曲空间丰富且自然发生。在项目中获得的理论见解可能适用于这些领域的具体问题。

AbstractAward: DMS-0503864Principal Investigator: Igor BelegradekThis project is concerned with the topology of curved Riemannianmanifolds. One objective is to study collapsed manifolds, i.e.manifolds that appear lower dimensional at a certain small butdefinite scale. Specifically, it is planned to investigate almostnonnegatively curved manifolds, collapsed manifolds of boundednonnegative curvature, and cusp cross-sections of finite volumenonpositively curved manifolds. In a somewhat different directionit is proposed to study negatively pinched manifolds, includingpinching estimates, rigidity in case of optimal pinching,finiteness theorems forced by assumptions on pinching, asymptoticgeometry of horospheres, and negatively pinched groups. Theproject continues earlier work of Belegradek and collaborators,and especially the recent results with Kapovitch ondiffeomorphism classification of negatively pinched manifoldswith amenable fundamental groups (that settled a problem ofBowditch), and on optimal pinching estimates for such manifolds(that answered a question of Gromov).While the project mostly deals with intrinsic development ofglobal Riemannian geometry, it is also to some extent motivatedby Sciences and Engineering, where curved spaces are plentifuland occur naturally. The theoretical insights obtained in theproject may be applicable to concrete problems in those fields.