Subharmonic Functions, Conformal Metrics, and CAT(0)

Abstract

We present an analytical proof that certain natural metric planar universal covers are Hadamard metric spaces. In particular if $\rho=\varphi\circ u$ where $u$ is locally Lipschitz and subharmonic in $\Omega$, $\varphi$ is positive and increasing on an interval containing $u(\Omega)$ with $\log\varphi$ convex, and if the metric space $(\Omega,\rho(z)|dz|)$ is complete, then it has universal cover $(\tilde{\Omega},\tilde{d})$ which is a Hadamard space for which geodesics have Lipschitz continuous first derivatives.