Alexandrov's Geometry and Applications

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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.