Subharmonic functions, conformal metrics, and CAT(0)

We present an analytical proof that certain natural metric universal covers are Hadamard metric spaces. If ρ d s induces a complete distance d on a plane domain Ω , and ρ = φ ∘ u where u is (locally Lipschitz and) subharmonic in Ω , φ is positive and increasing on an interval containing u ( Ω ) with log φ convex, then ( Ω , d ) has a universal cover ( Ω ~ , d ~ ) which is a Hadamard metric space (with geodesics that have Lipschitz continuous first derivatives).