Alexandrov's Geometry and Applications

Abstract DMS-0103957:

The project contains the following main topics:

1. Approximation of Riemannian

manifolds by polyhedral metrics and generalization of Alexandrov embedding

theorem. The PI has proved that approximability by polyhedral metrics

implies a new kind of curvature bound which appears naturally but has never been

considered 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 isometric

to a convex surface in a Euclidean space). This brings us back to a more

geometrical point of view on Riemannian Geometry, according to which

``interesting'' curvature bounds should arise from properties of

embeddings of manifolds into Euclidean space. This circle of ideas also

gives a new approach to the old conjecture that every simply connected

Riemannian manifold with positive curvature operator is diffeomorphic to a

sphere;

2. Collapsing with lower curvature bound. The general goal of this topic

is to understand how collapsing with lower curvature and diameter bounds

happens and, in the very best case, to construct a structure analogous to

the one obtained by Cheeger-Fukaya-Gromov for the case of bounded

curvature.

3. Gate spaces. Gate spaces are related to a circle of problems arising

from the question of K.Grove of whether there is an Alexandrov space that

has two different smoothings into Riemannian manifolds of the same

dimension and lower curvature bound;

4. Applications of megafolds to collapsing with bounded curvature and to

Ricci flow. Megafolds are a generalization of Riemannian manifolds and

orbifolds that has already proved it usefulness for collapsing with

bounded curvature; in particular, they were used by the PI to prove, in

coloboration with W.Tuschmann, the main part of the Klingenberg-Sakai

Conjecture. Such a collapsing also arises naturally from the rescaling of

the Ricci flow; it can be used to construct singularity models for the

Ricci flow with no injectivity radius estimates;

5. Theory of Alexandrov spaces. Alexandrov spaces appear naturally as

limits of Riemannian manifolds with lower curvature bound. Most geometric

results which are true for Riemaninan manifolds with lower curvature bound

are also true for Alexandrov spaces; however, there are several such

results that cannot be generalized. For example, it is not known whether a

convex hypersurface in an Alexandrov space is also an Alexandrov

space. Such problems are mostly due to the lack of local analysis, and

that is what the PI proposes to study.

Riemannian manifold, which could be considered as a simplified version of

space-time, is a way too complicated object. The first topic in this

proposal is aimed at studing Riemanian manifolds by means of approximation

by simpler objects. These objects are polyhedral spaces, i.e. spaces glued

of Euclidean polyhedra. The other topics considers a different approach to

studing Riemannian manifolds. It is based on considering extremal metrics,

in an appropriate sense, for example how Riemannian manifolds collapse to

lower dimenssional objects. This method makes possible to get new results

in the main stream direction of Riemannian geometry: how to make

conclusions about global structure of space basing on local properties.