Subharmonic functions, conformal metrics, and CAT(0)

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