It is a well known fact that locally CAT(-1) metrics on surfaces can be approximated by hyperbolic polyhedral metrics with cone singularities: roughly speaking you pick a geodesic triangulation of the surface and replace each triangle with an hyperbolic one.
Can we do better? More precisely, given a locally CAT(-1) distance $d$ on a closed surface, can we find a sequence of smooth metrics with curvature less than -1 such that the induced distance corverges to $d$? Or equivalently, can a polyhedral hyperbolic metric with cone angles bigger than $2π$ be approximated by a sequence of smooth metrics?