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 r-close to a ball in any splitting space R k+1 × Z. We show that there exists C(n, ) > 0 and β(n, ) > 0, independent of the volume, so that for any disjoint collection B r i (x i ) :This answers an open question in [8]. We also show that the k-singular set, 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 ⊆ S 1 and > 0 a space Y ∈ Alex 3 (0) with S 1 (Y) = φ(T ), where φ : S 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.

Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds

“…Example 2.14 (Li-Na [9]). For k ≤ n − 2 there exists a limit M n i , g i , p i → (X, d, p) where sec i ≥ 0 and Vol(B…”

“…Let X = C S 1 r be a metric cone over a circle of radius r < 1, thus one can picture X as an ice cream cone. We can smooth X to nonnegatively curved manifolds M i by rounding off the cone tip, see [9], and by Gauss-Bonnet one has R i dv i → 2π(1 − r)δ 0 , where δ 0 is the dirac delta measure at the cone point.…”

Gluing of multiple Alexandrov spaces

“…We claim that if z / ∈ C, then there exists r 0 > 0 so that g r (z) < 2 3 for every 0 < r < r 0 . Given δ > 0 small, by [LN19], there exists r 0 > 0 so that d GH (B s (z), B s (z * )) < δs for every 0 < s < 10r 0 , where z * ∈ T z (X 1 ) is the cone point. Let 0 < r < r 0 and ρ = g r (z) > 0.…”

“…where p * ∈ T p (X) is the cone point. See [LN19] for the existence of such an r. Note that ∂Σ p = {ξ, η} is a set of two points. Consider the gradient exponential map g exp p (tξ), 0 < t ≤ r. Let A ξ (r) = {g exp p (tξ) : t ∈ (0, r]} be the image of g exp p and A η (r) be defined similarly.…”

“…It was asked wether there are non-manifold designated methods to construct or confirm "non-trivial" Alexandrov spaces. Related work can be found in [LN19], where for every 1 ≤ k ≤ n − 2, Li and Naber construct Alexandrov spaces that are bi-Lipschitz to n-spheres and the (k, ǫ)-singular sets are k-dimensional fat Cantor sets. These examples can be viewed as Alexandrov spaces with C 1 -nontrivial metrics.…”