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  "body": "\n## Definition\n\nLet $X$ be a [[Riemann surface|5K0N8QMV6Q]] and let $U \\subseteq X$ be an open set.\nA function $u \\colon U \\to \\R$ is *harmonic* if for every holomorphic chart $\\phi \\colon V \\to \\phi(V) \\subseteq \\C$ of $X$, the function $u \\circ \\phi^{-1}$ is [[harmonic|0DEB0JMN0W]] on the open set $\\phi(U \\cap V)$.\n\n## Remarks\n\nIt suffices to check the condition for the charts of one atlas.\nIndeed, transition maps are holomorphic, and [[harmonicity is preserved by holomorphic maps|1VGB63K5HS]].\nEquivalently, $u$ is harmonic if each point of $U$ has a neighbourhood on which $u = \\operatorname{Re} f$ for a [[holomorphic|0NAHXR1FRH]] function $f$.\nExample: on $X = \\C \\setminus \\set{0}$ with the chart $z \\mapsto z$, the function $u(z) = \\log\\abs{z}$ is harmonic, since $\\Delta \\log\\abs{z} = 0$ there.\nNon-example: $u(z) = \\abs{z}^2$ on $X = \\C$ is not harmonic, since in the chart $z \\mapsto z$ its Laplacian is $4$.\nOn the [[Riemann sphere|0J78KBPFZB]], or on any compact Riemann surface, the maximum principle forces each harmonic function on the whole surface to be constant.\nThe Laplacian of $u$ depends on the chart, but the condition $\\Delta(u \\circ \\phi^{-1}) = 0$ does not; in coordinate-free form it reads $\\partial \\bar\\partial u = 0$.\nSubharmonic and superharmonic functions on $X$ are defined in the same way, chart by chart.\n",
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