On CAT($\kappa$) surfaces

Abstract

We study the properties of $\text{CAT}(\kappa)$ surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the $\text{CAT}(\kappa)$ condition locally. The main facts about $\text{CAT}(\kappa)$ 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 $\text{CAT}(\kappa)$ surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of $\text{CAT}(\kappa)$ surfaces. We also show that $\text{CAT}(\kappa)$ surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most $\kappa$. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces.