Quantitative Estimates on the Singular Sets of Alexandrov Spaces

Abstract

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