  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.
