Cohomogeneity one Alexandrov spaces

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric T ⁿ⁻¹ action. In contrast to the one- and two-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of \mathbbRP² \mathbb{R}{P^2} , which is not a manifold.     

A glance at three-dimensional Alexandrov spaces

We discuss the topology and geometry of closed Alexandrov spaces of dimension three.     

On the Betti numbers of Alexandrov spaces

In this note we show that the total Betti number of an Alexandrov space is bounded above by a constant depending only on the dimension, the diameter and the lower bound on curvature.Research supported in part by a Grant from Department of Mathematics, University of South Carolina.     

Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces

We prove the compactness of the imbedding of the Sobolev space into for any relatively compact open subset of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally H?lder continuous heat kernel. Received: 27 December 1999 / in final form: 1 February 2000 / Published online: 4 May 2001     

Covering radii and paving diameters of Alexandrov spaces

Covering radii and paving diameters are defined, and the borderline case when cov_k X = π/2, k = 1,…,n + 1 and pav_k X = π/2, k = 1,…,n + 1 is studied (curv X ≥1, dim X = n).     

On the spread of positively curved Alexandrov spaces

It was proved by F. Wilhelm that Gromov’s filling radius of closed positively curved manifolds with a uniform lower bound on sectional curvature attains the maximum with the round sphere. Recently the author proved that this is also the case for closed finite-dimensional Alexandrov spaces with a positive lower curvature bound. These were proved as a corollary of a comparison theorem for the invariant called spread of those spaces. In this paper, we extend the latter result to infinite-dimensional Alexandrov spaces.     

Quasi-convex subsets in Alexandrov spaces with lower curvature bound

We introduce quasi-convex subsets in Alexandrov spaces with lower curvature bound, which include not only all closed convex subsets without boundary but also all extremal subsets. Moreover, we explore several essential properties of such kind of subsets including a generalized Liberman theorem. It turns out that the quasi-convex subset is a nice and fundamental concept to illustrate the similarities and differences between Riemannian manifolds and Alexandrov spaces with lower curvature bound.     

Existence and uniqueness of optimal maps on Alexandrov spaces

The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.     

On the filling radius of positively curved Alexandrov spaces

It was shown by F. Wilhelm that Gromov’s filling radius of any positively curved closed Riemannian manifolds are less than that of the round sphere unless they are isometric to each other. In this short paper, we adapt his proof to see that the same is true for any positively curved closed Alexandrov spaces as well.     

On tangent cones of Alexandrov spaces with curvature bounded below

The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric. Received: 20 October 1999 / Revised version: 8 May 2000     

A rigidity theorem in Alexandrov spaces with lower curvature bound

Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang–Schroeder–Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces which possibly have infinite dimension and are not supposed to be locally compact.     

A volume convergence theorem for Alexandrov spaces with curvature bounded above

We obtain a volume convergence theorem for Alexandrov spaces with curvature bounded above with respect to the Gromov-Hausdorff distance. As one of the main tools proving this, we construct an almost isometry between Alexandrov spaces with curvature bounded above, with weak singularities, which are close to each other. Furthermore, as an application of our researches of convergence phenomena, for given positive integer , we prove that, if a compact, geodesically complete, n-dimensional CAT(1)-space has the volume sufficiently close to that of the unit n-sphere, then it is bi-Lipschitz homeomorphic to the unit n-sphere. Received: 30 January 2001; in final form: 30 October 2001 / Published online: 4 April 2002     

Lipschitz Gromov-Hausdorff appr oximat ions to two-dimensional closed piecewise flat Alexandrov spaces

We show that a closed piecewise fiat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdorff topology, there is a Lipschitz Gromov-Hausdorff approximation.