On CAT($\kappa$) surfaces

TL;DR

This paper rigorously establishes that CAT(κ) surfaces, which are length metric surfaces with curvature bounded above, have bounded curvature and can be approximated by smooth Riemannian surfaces, clarifying their mathematical properties.

Contribution

Provides a complete proof that CAT(κ) surfaces have bounded curvature and introduces explicit smoothing formulas for polyhedral models, filling gaps in the existing folklore.

Findings

CAT(κ) surfaces have bounded (integral) curvature.

They can be approximated by smooth Riemannian surfaces.

Explicit smoothing formulas for polyhedral vertices are provided.

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.