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 $(k,\epsilon )$-singular set ${\mathcal{S}}_{\epsilon ,r}^k(X)$ 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 $\{B_{r_i}(x_i):x_i\in {\mathcal{S}}_{\epsilon ,\beta r_i}^k(X)\cap B_1,r_i\le 1\}$, the packing estimate $\sum r_i^k\le C$ holds. Consequently, we obtain the Hausdorff measure estimates ${\mathcal{H}}^k({\mathcal{S}}_{\epsilon }^k(X)\cap B_1)\le C$ and ${\mathcal{H}}^n(B_r({\mathcal{S}}_{\epsilon ,r}^k(X))\cap B_1(p))\le C{r}^{n-k}$. This answers an open question asked by Kapovitch and Lytchak. We also show that the $k$-singular set ${\mathcal{S}}^k(X)=\underset{\epsilon >0}{\cup }\left(\underset{r>0}{\cap }{\mathcal{S}}_{\epsilon ,r}^k\right)$ 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 ${\mathcal{S}}_{\epsilon }^1(Y)=\varphi (T)$, where $\varphi :{\mathbb{S}}^1\to Y$ 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.

28 pages