Quantitative Estimates on the Singular Sets of Alexandrov Spaces

TL;DR

This paper establishes quantitative estimates on the size and structure of singular sets in Alexandrov spaces with curvature bounded below, proving rectifiability and providing explicit bounds on their measure and packing properties.

Contribution

It introduces new packing and measure estimates for singular sets in Alexandrov spaces, confirming their rectifiability and answering an open question in the field.

Findings

Packing estimates for singular sets independent of volume

Hausdorff measure bounds for singular sets

Construction of examples demonstrating sharpness of rectifiability

Abstract

Let $X\in\text{Alex}\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\ge -1$. Let the $r$-scale $(k,\epsilon)$-singular set $\mathcal S^k_{\epsilon,\,r}(X)$ be the collection of $x\in X$ so that $B_r(x)$ is not $\epsilon r$-close to a ball in any splitting space $\mathbb R^{k+1}\times Z$. We show that there exists $C(n,\epsilon)>0$ and $\beta(n,\epsilon)>0$, independent of the volume, so that for any disjoint collection $\big\{B_{r_i}(x_i):x_i\in \mathcal S_{\epsilon,\,\beta r_i}^k(X)\cap B_1, \,r_i\le 1\big\}$, the packing estimate $\sum r_i^k\le C$ holds. Consequently, we obtain the Hausdorff measure estimates $\mathcal H^k(\mathcal S^k_\epsilon(X)\cap B_1)\le C$ and $\mathcal H^n\big(B_r (\mathcal S^k_{\epsilon,\,r}(X))\cap B_1(p)\big)\leq C\,r^{n-k}$. This answers an open question asked by Kapovitch and Lytchak. We also show that the $k$-singular set $\mathcal S^k(X)=\underset{\epsilon>0}\cup\left(\underset{r>0}\cap\mathcal S^k_{\epsilon,\,r}\right)$ 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 $T\subseteq \mathbb S^1$ and $\epsilon>0$ a space $Y\in\text{Alex}^3(0)$ with $\mathcal S^{1}_\epsilon(Y)=\phi(T)$, where $\phi\colon\mathbb S^1\to Y$ 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.