Alexandrov's Geometry and Applications

DMS-0103957：本项目包括以下主要内容：1.黎曼流形的多面体度量逼近和Alexandrov嵌入定理的推广。PI证明了多面体度量的可逼近性给出了一种新的曲率界，这种界在几何中是自然出现的，但在几何中却从未被考虑过。这个证明本身很有趣，因为它使用了一个与Alexandrov嵌入定理非常相似的引理(该定理指出，2-球面上的任何正弯曲度量都等距于欧氏空间中的凸曲面)。这把我们带回到了关于黎曼几何的一个更多的计量学观点，根据这个观点，“有趣的”曲率界应该由欧几里德空间中流形的嵌入的性质产生。这一思想圈还给出了一种新的方法来证明每个具有正曲率算子的单连通黎曼流形都是非球面的微分同胚；2.以较低的曲率下界进行折叠。本主题的总体目标是了解较低曲率和直径的坍塌是如何有界的，并且在最好的情况下，构造一个类似于Cheeger-Fukaya-Gromov在有界曲率的情况下得到的结构。星际空间。门空间与K.Grove关于是否存在两个不同光滑到相同维度和曲率下界的黎曼流形的Alexandrov空间的问题有关；4.大流形在有界曲率折叠和Ricci流中的应用。Megafold是黎曼流形的推广，已经证明了它对于有界曲率的折叠是有用的；特别是，PI用它们来证明Klingenberg-Sakai猜想的主要部分W.Tuschmann。这种塌缩也是由于Ricci流的重新标度而自然产生的；它可以用来构造Ricci流的奇点模型，而不需要内射半径估计；5.亚历山大空间理论。Alexandrov空间自然成为曲率下界的黎曼流形的极限。大多数几何结果对于曲率下界的Riemaninan流形是正确的，对于Alexandrov空间也是正确的；然而，有几个这样的结果不能推广。例如，Alexandrov空间中的凸超曲面是否也是Alexandrov空间就不得而知了。这些问题主要是由于缺乏局域分析，这也是PI提出要研究的问题。黎曼流形可以被认为是时空的简化版本，但它是一种过于复杂的对象。本文的第一个主题是研究黎曼流形的更简单对象的逼近。这些对象是多面体空间，即欧几里得多面体的粘性空间。其他主题考虑一种不同的方法来研究黎曼流形。它是基于在适当的意义上考虑极值度量的，例如，黎曼流形如何塌陷到低维对象。这种方法使得在黎曼几何的主流方向：如何根据局部性质对空间的整体结构作出结论成为可能。

Abstract DMS-0103957:The project contains the following main topics: 1. Approximation of Riemannian manifolds by polyhedral metrics and generalization of Alexandrov embeddingtheorem. The PI has proved that approximability by polyhedral metricsimplies a new kind of curvature bound which appears naturally but has never beenconsidered in geometry before. The proof is interesting in its own right,as it uses a Lemma closely resembling the Alexandrov embedding theorem(which states that any positively curved metric on a 2-sphere is isometricto a convex surface in a Euclidean space). This brings us back to a moregeometrical point of view on Riemannian Geometry, according to which``interesting'' curvature bounds should arise from properties ofembeddings of manifolds into Euclidean space. This circle of ideas alsogives a new approach to the old conjecture that every simply connectedRiemannian manifold with positive curvature operator is diffeomorphic to asphere; 2. Collapsing with lower curvature bound. The general goal of this topicis to understand how collapsing with lower curvature and diameter boundshappens and, in the very best case, to construct a structure analogous tothe one obtained by Cheeger-Fukaya-Gromov for the case of boundedcurvature.3. Gate spaces. Gate spaces are related to a circle of problems arisingfrom the question of K.Grove of whether there is an Alexandrov space thathas two different smoothings into Riemannian manifolds of the samedimension and lower curvature bound;4. Applications of megafolds to collapsing with bounded curvature and to Ricci flow. Megafolds are a generalization of Riemannian manifolds andorbifolds that has already proved it usefulness for collapsing withbounded curvature; in particular, they were used by the PI to prove, incoloboration with W.Tuschmann, the main part of the Klingenberg-SakaiConjecture. Such a collapsing also arises naturally from the rescaling ofthe Ricci flow; it can be used to construct singularity models for theRicci flow with no injectivity radius estimates; 5. Theory of Alexandrov spaces. Alexandrov spaces appear naturally aslimits of Riemannian manifolds with lower curvature bound. Most geometricresults which are true for Riemaninan manifolds with lower curvature boundare also true for Alexandrov spaces; however, there are several suchresults that cannot be generalized. For example, it is not known whether aconvex hypersurface in an Alexandrov space is also an Alexandrovspace. Such problems are mostly due to the lack of local analysis, andthat is what the PI proposes to study.Riemannian manifold, which could be considered as a simplified version ofspace-time, is a way too complicated object. The first topic in thisproposal is aimed at studing Riemanian manifolds by means of approximationby simpler objects. These objects are polyhedral spaces, i.e. spaces gluedof Euclidean polyhedra. The other topics considers a different approach tostuding Riemannian manifolds. It is based on considering extremal metrics,in an appropriate sense, for example how Riemannian manifolds collapse tolower dimenssional objects. This method makes possible to get new resultsin the main stream direction of Riemannian geometry: how to makeconclusions about global structure of space basing on local properties.