Collapsing, pinching, and curvature bounds

折叠、收缩和曲率边界

• 批准号: 0503864
• 负责人: Igor Belegradek
• 金额: --
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503864&HistoricalAwards=false
• 关键词: Collapsing; pinching; curvature; bounds

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.

摘要奖：DMS-0503864主要研究人员：伊戈尔·贝勒格雷这个项目是关于弯曲黎曼流形的拓扑。一个目标是研究折叠流形，即在某个小的但确定的尺度上看起来较低维度的流形。具体地说，它计划研究几乎大多数非负弯曲流形，有界非负曲率的折叠流形，以及有限体积非正弯曲流形的尖点截面。在一个稍有不同的方向上，建议研究负收缩流形，包括收缩估计，最优收缩情况下的刚性，由收缩假设所强制的有限定理，星球的渐近几何，以及负收缩群。该项目延续了Belegradek及其合作者早期的工作，特别是最近与Kapovitch在具有服从基本群的负收缩流形的微分同态分类(解决了Bowditch的一个问题)以及此类流形的最优Pinch估计(回答了Gromov的一个问题)方面的结果。虽然该项目主要涉及全球黎曼几何的内在发展，但它在某种程度上也受到科学和工程的启发，其中弯曲空间是丰富的并且自然发生的。该项目所获得的理论见解可能适用于这些领域的具体问题。