Subharmonic Functions, Conformal Metrics, and CAT(0)

TL;DR

This paper provides an analytical proof that certain metric planar universal covers are Hadamard spaces, focusing on metrics derived from subharmonic functions composed with increasing convex functions, ensuring non-positive curvature properties.

Contribution

The paper introduces a new analytical approach to establish that specific metric spaces constructed from subharmonic functions are Hadamard spaces, expanding understanding of their geometric structure.

Findings

Universal covers are Hadamard spaces with Lipschitz continuous geodesics.

Metrics derived from subharmonic functions satisfy CAT(0) conditions.

Complete metric spaces with these properties have non-positive curvature.

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.