Curvature and Metric Geometry

本项目主要研究各种度量度量空间的解析、几何和拓扑性质，这些空间虽然比光滑黎曼流形奇异得多，但却具有足够好的性质，可以产生丰富而有趣的理论。具体地说，我们考虑具有确定的下（或双面）里奇曲率界的黎曼流形（包括爱因斯坦流形）序列（可能坍缩）的测量的Gromov-Haudorff极限，具有确定的双面截面曲率界的流形和度量度量空间（本质上可能是分形的）的相应极限，这些空间满足度量和庞加莱不等式上的加倍条件。一个中心问题是证明“部分正则性定理”。这样的定理断言底层空间是正则的（在适当的意义上）在一个测度为零或甚至是某个确定的正余维的集合上。在整个数学和物理学中，人们发现“规律性”和“奇点”这两个概念之间存在张力。例如，一个“典型”对象（完全随机选择的对象）很可能是非常不规则的（甚至是单一的）。另一方面，最基本的几何形状，比如圆球，表现出对称、均匀和平滑。在任何情况下，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非常重要。这正是我们项目的重点。我们考虑一些非常一般但非常自然的几何物体类别的问题。

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.