Quantitative Estimates on the Singular Sets of Alexandrov Spaces

Abstract

Let X ∈ Alex n (−1) be an n-dimensional Alexandrov space with curvature ≥ −1. Let the r-scale (k,)-singular set S k , r (X) be the collection of x ∈ X so that B r (x) is not rclose to a ball in any splitting space ℝ k+1 × Z. We show that there exists C(n,) > 0 and (n,) > 0 , independent of the volume, so that for any disjoint collection B r i (x i) ∶ x i ∈ S k , r i (X) ∩ B 1 , r i ≤ 1 , the packing estimate ∑ r k i ≤ C holds. Consequently, we obtain the Hausdorff measure estimates H k (S k (X) ∩ B 1) ≤ C and H n B r (S k , r (X)) ∩ B 1 (p) ≤ C r n−k. This answers an open question in Kapovitch et al. (Metric-measure boundary and geodesic flow on Alexandrov spaces. arXiv : 1705.04767 (2017)). We also show that the k-singular set S k (X) = ⋃ >0 � ⋂ r>0 S k , r � is k-rectifiable and construct examples to show that such a structure is sharp. For instance, in the k = 1 case we can build for any closed set T ⊆ 1 and > 0 a space Y ∈ Alex 3 (0) with S 1 (Y) = (T) , where ∶ 1 → Y is a bi-Lipschitz embedding. Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable, 1-Cantor set with positive 1-Hausdorff measure.

Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below

“…The structural results above are actually sharp. In Example 3.2 we will explain a construction from [LiNa17] of a noncollapsed limit space X n such that…”

“…One of the primary results of this paper is to show that the kth-stratum S k \ S k−1 of the singular set is k-rectifiable. The following example from [LiNa17] shows that this statement is sharp in the sense that their need not exist any points in the singular set S in a neighborhood of which S is a manifold.…”

“…We will briefly explain the example above from [LiNa17] for the case, 0 ≤ s ≤ 1. Higher dimensional examples are built in an analogous manner.…”

Boundary regularity and stability for spaces with Ricci bounded below

“…• As pointed out in [29,Remark 1.11,Example 3.2] based on [66], there is an example of N -dimensional Alexandrov space such that the singular set S N −2 is a Cantor set, and in particular no point has a neighbourhood in which S N −2 is topologically a manifold.…”

“…In particular, most of the results of the present paper follow from the Alexandrov theory if N = 2. (v) Relying on [66,Corollary 1.4] instead of [29] it is possible to prove that Theorem 1.5 holds also when (X, d, H N ) is an Alexandrov space with curvature bounded from below.…”