Let $(X,d)$ be compact metric space of curvature greater than $-1$ (in the sense of comparison triangles), assume that its Hausdorff dimension is $2$. Then a result of Perelman says that $X$ is a 2-dimensional manifold.

## Claim

$(X,d)$ can be Gromov-Hausdorff approximated by a sequence of Riemannian surfaces $(M_i,g_i)$ such that $\int_{M_i}|K_{g_i}|dv_{g_i}$ is bounded.

To see this :

A version of the Gauss-Bonnet Theorem holds (Machigashira, The Gaussian curvature of Alexandrov surfaces), which implies that $(X,d)$ is an Alexandrov surface with bounded

*integral*curvature.Any such surface can be approximated by smooth Riemannian surfaces with bounded integral curvature. See Reshetnyak, Geometry IV, Encyclopaedia of Mathematical Sciences.

## Question

Is any compact Alexandrov surface of curvature greater than $-1$ approximated by a sequence of smooth compact Riemannian surfaces with curvature bounded from below (by -1, or something else if this helps) ?

Maybe this is classic but I didn't find explicit results of this kind.

Thanks.