Quantitative estimates on the $C^2$-singular sets in Alexandrov spaces

The total disaster may be controllable if not preventable. We will explore this phenomenon for singularities in metric spaces. A point in an $n$-dimensional Alexandrov space is called regular if its tangent cone is isometric to $\mathbb R^n$. Examples show that not every regular point is smooth, and the non-smooth points, away from the boundary, can have co-dimension 1. In this paper, we define a non-negative function $\mathcal K(x)$, which quantitatively measures the extent of the point $x$ from being $C^2$. The so-called $C^2$-singular points are identified as the set where $\mathcal K>0$. We show that $\int_{B_r(p)} \mathcal K(x)\, \operatorname d\mathcal H^{n-1}\le c(n,\kappa,\nu)r^{n-2}$ for any $n$-dimensional Alexandrov space $(X,p)$ with curv $\ge \kappa$ and $\operatorname{Vol}\left(B_1(p)\right)\ge\nu>0$. This leads to the Hausdorff dimension estimate $\dim_\mathcal H\{\mathcal K>0\}\le n-1$, and the quantitative Hausdorff measure estimate $\mathcal H^{n-1}\left(\{\mathcal K>\epsilon\}\cap B_r(p)\right)\le \epsilon^{-1}\cdot c(n,\nu)r^{n-2}$. These results also make progress on Naber's conjecture on the convergence of curvature measures. The measure $\mathcal K(x)\, \operatorname d\mathcal H^{n-1}$ on Alexandrov spaces can be viewed as the counterpart of the curvature measure $scal \,\operatorname d {vol}_{g}$ on smooth manifolds. We also show that if $n$-dimensional Alexandrov spaces $X_i$ Gromov-Hausdorff converge to a smooth manifold with no boundary without collapsing, then $\mathcal K_i\, \operatorname d\mathcal H^{n-1}\to 0$ as a measure.