We present an analytical proof that certain natural metric universal covers are Hadamard metric spaces. If $$\rho \,\mathrm{{d}}s$$ induces a complete distance d on a plane domain $$\Omega $$ , and $$\rho =\varphi \circ u$$ where u is (locally Lipschitz and) subharmonic in $$\Omega $$ , $$\varphi $$ is positive and increasing on an interval containing $$u(\Omega )$$ with $$\log \varphi $$ convex, then $$(\Omega ,d)$$ has a universal cover $$({\tilde{\Omega }},{\tilde{d}})$$ which is a Hadamard metric space (with geodesics that have Lipschitz continuous first derivatives).

Subharmonic functions, conformal metrics, and CAT(0)

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