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  "body": "\n## Definition\n\nLet $X$ be a connected [[Riemann surface|5K0N8QMV6Q]].\nA function on $X$ is subharmonic if it is a [[subharmonic function|Z1MHMGFNFW]] in every holomorphic chart, as for [[harmonic functions on Riemann surfaces|P214QJ0S3B]].\n\n1. $X$ is *hyperbolic* if there is a nonconstant subharmonic function $u$ on $X$ with $u \u003c 0$ on $X$.\n2. $X$ is *parabolic* if $X$ is not compact and not hyperbolic.\n\n## Remarks\n\nEquivalently, $X$ is hyperbolic if and only if it has a Green's function.\nThis is the smallest positive harmonic function on $X \\setminus \\set{a}$ with a logarithmic pole at $a$.\nA compact Riemann surface is never hyperbolic, because a subharmonic function on it attains its maximum and is therefore constant; such surfaces are sometimes called elliptic.\nExample: the unit disc $\\mathbb{D}$ is hyperbolic, since $u(z) = \\abs{z}^2 - 1$ is subharmonic, nonconstant and negative on $\\mathbb{D}$.\nNon-example: the plane $\\C$ is not hyperbolic.\nBy the subharmonic form of [[Liouville's theorem|T12FSF2J2V]], a subharmonic function on $\\C$ that is bounded above is constant.\nAs $\\C$ is not compact, it is parabolic.\nA domain $\\Omega \\subseteq \\C$ is hyperbolic if and only if its boundary is not a [[polar set|C95QJ7YH5T]].\nIts Green's function is then the classical [[Green's function|M36SD56MWQ]] of $\\Omega$.\nThis notion differs from the one in the uniformization theorem, which classifies $X$ by its [[universal cover|Y1YPE6PM52]].\nFor example, $\\C \\setminus \\set{0, 1}$ has the disc as universal cover, but it is parabolic, because a subharmonic function bounded above extends across the two punctures to all of $\\C$.\n",
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