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.