On CAT(κ) surfaces

We study the properties of CAT(κ) surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the CAT(κ) condition locally. The main facts about CAT(κ) surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation. We provide a complete proof that CAT(κ) surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of CAT(κ) surfaces. We also show that CAT(κ) surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most κ. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces.