A volume convergence theorem for Alexandrov spaces with curvature bounded above

Abstract

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 n, 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.

Spaces of Curvature Bounded Above

“…For simplicity, we give a qualitative version. For the proof see [49, §3], a quantitative version can be found in [88]. This theorem is similar to [47,Theorem 5.4] for CBB-spaces, and the argument follows a similar line of reasoning by proving the existence of a distance frame or strainer and studying the associated distance map into R n .…”

“…It is proven in [88] that then H n (X) ≥ vol S n for the unit sphere S n ⊂ R n+1 . Moreover, if in addition H n (X) < vol S n + ε n for some ε n > 0 depending only on n ≥ 1, then X is bi-Lipschitz homeomorphic to S n with bi-Lipschitz constants close to 1.…”

“…For applications to structure results for CBA-metrics on 2-polyhedra see [49] and sect. 10, for applications to a volume convergence theorem see [88] and the end of sect. 5.…”

Open map theorem for metric spaces

“…such that for each w ∈ Y The last two results were proved and used in several particular situations in [BGP92,Nag02,Per94] and some other papers. The following are the rigid versions of the above results.…”

Collapsing manifolds with boundary

“…However, neither requires geodesic extendibility of the approximating space. For spaces of curvature bounded above, there seem to be known only two fibering theorems: Theorem 2.2.1, specific to manifolds (and not involving Gromov-Hausdorff distance per se), and Nagano's Theorem in [13], concerning geodesically extendible Alexandrov spaces of curvature bounded above (with also a mild condition on geodesic branching, and a uniform CAT k radius bound). The latter theorem requires geodesic extendibility of both the fixed (limit) space and the approximating space, and concludes the existence of a bi-Lipschitz homeomorphism between them.…”