Subharmonic Functions, Conformal Metrics, and CAT(0)

David A. Herron; Gaven J. Martin

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arXiv · 2011.14456

Subharmonic Functions, Conformal Metrics, and CAT(0)

David A. Herron·Gaven J. Martin

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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.

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David A. Herron, Gaven J. Martin. 2020-11-29. Subharmonic Functions, Conformal Metrics, and CAT(0). https://arxiv.org/abs/2011.14456

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