On CAT($κ$) surfaces

TL;DR

The paper clarifies that CAT($\kappa$) surfaces in two dimensions have bounded curvature and can be approximated by smooth Riemannian surfaces with Gaussian curvature at most $\kappa$, tying CAT geometry to the classical theory of surfaces of bounded curvature. A central tool is the vertex-edge triangulation framework, used to decompose and refine triangulations while controlling excess and model-area, leading to the key inequality $\delta(T) \le \kappa|T|$. The authors provide a constructive smoothing procedure at vertices to obtain uniform limits by smooth metrics, enabling a precise link between polyhedral and smooth models. These results situate CAT($\kappa$) surfaces within a broader network of Alexandrov geometry, bi-Lipschitz uniformization results, and isoperimetric-type properties, with implications for area notions and uniformization theorems. Overall, the work offers complete, transparent proofs of foundational facts and a practical smoothing method that clarifies the landscape of two-dimensional CAT spaces.

Abstract

We study the properties of $\text{CAT}(κ)$ surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the $\text{CAT}(κ)$ condition locally. The main facts about $\text{CAT}(κ)$ 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 $\text{CAT}(κ)$ surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of $\text{CAT}(κ)$ surfaces. We also show that $\text{CAT}(κ)$ 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.

Figures (3)

• Figure 4.1: A polygon as in \ref{['lemm:recognize_ve_partition']}
• Figure 4.2:
• Figure 4.3: Subdividing a $\mathop{\mathrm{CAT}}\nolimits(\kappa)$ triangle

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Theorem 2.4
• Lemma 3.1