Bounding Curvature Measure on Manifolds with Singularities

Let $X$ be an $n$-dimensional Alexandrov space with curvature $\ge -1$, and let $\eta >0$. Define ${\mathcal{S}}_{\eta }^k(X)$ as the set of $(k,\eta )$-singular points in $X$ whose tangent cones are $\eta$-away from splitting off ${\mathbb{R}}^{k+1}$ isometrically. For a point $p\in X$, assume that $M=B_2(p)\setminus ({\mathcal{S}}_{\eta }^{n-2}(X)\cup \mathrm{\partial }X)$ is a smooth manifold equipped with the Riemannian metric induced by $X$. We prove that the integral of the scalar curvature of $M$ over $B_1(p)$ is bounded from above by a constant depending only on $n$ and $\eta$. As a special case, this extends Petrunin’s bounded curvature integral result for complete manifolds with lower sectional curvature bound to the setting of open manifolds and smooth manifolds with boundary, provided that these manifolds are Alexandrov spaces.