Alexandrov's Geometry and Applications

摘要DMS-0103957：该项目包含以下主要主题：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简化了一种新的曲率界，这种曲率界自然出现，但在几何学中从未被考虑过。这个证明本身就很有趣，因为它使用了一个非常类似于亚历山德罗夫嵌入定理的引理（该定理指出，在一个2-球面上的任何正弯曲度量都等距于一个欧几里得空间中的凸曲面）。这使我们回到了黎曼几何的一个更几何的观点，根据这个观点，“有趣的”曲率界应该产生于流形嵌入欧几里得空间的性质。这一思想循环也给了一个新的方法，以旧的猜想，每一个单连通黎曼流形的正曲率算子是仿射到非球面; 2。以较低的曲率边界塌陷。本文的总体目标是理解具有较低曲率和直径边界的坍缩是如何形成的，并在最好的情况下，构造一个类似于Cheeger-Grosaya-Gromov在有界曲率情况下得到的结构.星门空间。 门空间与K.格罗夫关于是否存在一个Alexandrov空间的一系列问题有关，Alexandrov空间是两个不同的光滑空间，光滑空间是同一维的黎曼流形，具有相同的下曲率界;巨型体在有界曲率坍缩和Ricci流中的应用。 Megafolds是黎曼流形和orbifolds的推广，已经证明了它对有界曲率的坍缩是有用的;特别是，它们被PI用来证明，与W.Tuschmann合作，Klingenberg-Sakai猜想的主要部分。这种坍缩也自然地产生于Ricci流的重新标度;它可以用于构造没有注入半径估计的Ricci流的奇异性模型; 5.亚历山德罗夫空间理论 Alexandrov空间自然地表现为具有下曲率界的黎曼流形的极限。大多数几何结果对具有较低曲率边界的Riemaninan流形是正确的，对Alexandrov空间也是正确的;然而，有几个这样的结果不能推广。例如，不知道亚历山德罗夫空间中的凸超曲面是否也是亚历山德罗夫空间。这些问题主要是由于缺乏局部分析，而这正是PI提出要研究的。黎曼流形，可以被认为是时空的简化版本，是一个过于复杂的对象。第一个课题是用较简单的物体逼近的方法来研究黎曼流形。这些物体是多面体空间，即欧氏多面体的空间。其他主题考虑了一种不同的方法来研究黎曼流形。它是基于考虑极值度量，在适当的意义上，例如如何黎曼流形崩溃tolower dimensional对象。这一方法使黎曼几何主流方向上的新结果：如何根据局部性质作出空间整体结构的结论成为可能。

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.