Curvature and Metric Geometry

曲率和公制几何

• 批准号: 9803171
• 负责人: Jeff Cheeger
• 金额: $ 12.37万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803171&HistoricalAwards=false
• 关键词: Curvature; Metric; Geometry

Abstract Proposal: DMS-9803171 Principal Investigator: Jeff Cheeger The main thrust of this project is to study the analytic, geometric and topological properties of various classes of metric measure spaces, which while typically much more singular than smooth riemannian manifolds, have enough good properties to give rise to rich and interesting theories. Specifically, we consider measured Gromov-Haudorff limits of (possibly collapsing) sequences of riemannian manifolds with a definite lower (or two sided) Ricci curvature bound (including Einstein manifolds), corresponding limits of manifolds with a definite two sided sectional curvature bound and metric measure spaces (possibly fractal in nature) which satisfy a doubling condition on the measure and a Poincar'e inequality. A central concern is that of proving ``partial regularity theorems''. Such theorems assert that the underlying space is regular (in a suitable sense) off a set which is of measure zero or even of some definite positive codimension. Throughout mathematics and physics one finds a tension between the notions of ``regularity'' and ``singularity''. For instance, a ``typical'' object (one which is chosen completely at random) will, with high probability, be very irregular (or even singular). On the other hand, the most fundamental geometric shapes, say that of round sphere, exhibit symmetry, homgeneiety, and smoothness. In any event, in many (if not most) problems of interest, singularities are simply ``there'' and cannot be avoided (e.g. ``black holes''). In such cases, in becomes important to study the extent and structure of the singularities. This is precisely the focus of our project. We consider the problem for some very general but extremly natural classes of geometric objects.

摘要 提案：DMS-9803171主要研究者：Jeff Cheeger 这个项目的主旨是研究各类度量测度空间的分析、几何和拓扑性质，度量测度空间虽然比光滑黎曼流形奇异得多，但有足够好的性质来产生丰富有趣的理论。 具体地说，我们考虑的测量Gromov-Haudorff极限（可能崩溃）序列的黎曼流形与一个明确的下（或两侧）里奇曲率界（包括爱因斯坦流形），相应的极限流形与一个明确的两侧截面曲率界和度量测度空间（可能分形性质），满足一个加倍条件的措施和庞加莱不等式。一个中心问题是证明“部分正则性定理”。 这样的定理断言，基本空间是正则的（在适当的意义上）的一个集合，这是零的措施，甚至一些明确的正余维。 在整个数学和物理学中，人们发现“正则性”和“奇异性”的概念之间存在着紧张关系。 例如，一个“典型的”对象（一个完全随机选择的对象）很有可能是非常不规则的（甚至是奇异的）。 另一方面，最基本的几何形状，比如说圆球，表现出对称性、均匀性和光滑性。 无论如何，在许多（如果不是大多数）感兴趣的问题中，奇点只是“存在”并且无法避免（例如“黑洞”）。 在这种情况下，研究奇点的范围和结构就变得很重要。 这正是我们项目的重点。 我们考虑的问题，一些非常普遍的，但非常自然的类的几何对象。