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 ${\mathcal {S}}^k_{\epsilon ,\,r}(X)$ 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 ${\mathcal {H}}^n\big (B_r ({\mathcal {S}}^k_{\epsilon ,\,r}(X))\cap B_1(p)\big )\le C\,r^{n-k}$. 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 $\textstyle{\mathcal {S}}^k(X)=\bigcup_{\epsilon>0}\big(\bigcap_{r>0}{\mathcal {S}}^k_{\epsilon ,\,r}\big)$ 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 $\phi :\mathbb {S}^1\rightarrow Y$ 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.

Funding

Directorate for Mathematical and Physical Sciences(1809011); PSC-CUNY(61533-0049)