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 $\rho =\phi \circ u$ where $u$ is locally Lipschitz and subharmonic in $\Omega$, $\phi$ is positive and increasing on an interval containing $u(\Omega )$ with $\log \phi$ 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.

arXiv admin note: substantial text overlap with arXiv:2007.00782