Completions, branched covers, Artin groups and singularity theory

Understand

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry.

• Here curvature bounds are expressed in terms of the CAT(k) inequality.
• We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT(k) metric space for the completion to be CAT(k).
• We use this to prove that a branched cover of a complete Riemannian manifold is locally CAT(k) if and only if all tangent spaces are CAT(0) and the base has sectional curvature bounded above by k.