Alexandrov spaces with large volume growth

Abstract

Let (X, d) be an n-dimensional Alexandrov space whose Hausdorff measure H n satisfies a condition giving the metric measure space (X, d, H n ) a notion of having nonnegative Ricci curvature. We examine the influence of large volume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then (X, d, H n ) has finite topological type.

The Caffarelli-Kohn-Nirenberg Inequalities on Metric Measure Spaces

“…Let us denote by Alex n [κ] the class of n-dimensional Alexandrov space of curvature ≥ κ. In [18] see Theorem 3.2, the authors prove the following Theorem 5.7. For an integer n ≥ 2, let (X, d) ∈ Alex n [−κ 2 ], κ ∈ R be a complete non-compact Alexandrov space whose Hausdorff measure H n satisfies the BG(0, n) condition.…”

Shape of Alexandrov spaces with positive Ricci curvature

“…For the case of Alexandrov spaces, the Abresch-Gromoll inequality [GM,M,ZZ2] and the Bishop-Gromov volume comparison inequality [LV,O,S2] still hold, the remain is to show that there exists δ = δ(n) > 0 such that if V ol(M ) ≥ ω n − δ then M is a topological manifold, which follows from (a) By Lemma 4.1(i), given any ǫ > 0, there exists δ = δ(ǫ, n) > 0 such that if V ol(M ) ≥ ω n − δ then V ol(Σ x ) ≥ ω n−1 − ǫ for any x ∈ M ;…”