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  "body": "\n## Statement\n\nLet $u \\colon \\C \\to \\R$ be a [[harmonic function|0DEB0JMN0W]] on the whole plane.\nAssume that $u$ is bounded above or bounded below.\nThen $u$ is constant.\n\n## Remarks\n\nProof idea: $\\C$ is simply connected, so by [[the existence of harmonic conjugates|CVDGP7DKZ3]] there is an entire function $f$ with $\\operatorname{Re} f = u$.\nIf $u \\le M$ on $\\C$, then $g = e^{f}$ is entire with $\\abs{g} = e^{u} \\le e^{M}$.\nBy [[Liouville's theorem|0EHKVRF4J6]], $g$ is constant, so $f' e^{f} = g' = 0$, hence $f' = 0$ and $u$ is constant.\nIf $u$ is bounded below, apply this to $-u$.\nA one-sided bound is needed: $u(x + iy) = x$ is harmonic, nonconstant, and unbounded in both directions.\nThe theorem is the planar case of [[Liouville's theorem for harmonic functions|QK1KWYZNRD]] on $\\R^n$.\nIn the plane even more is true: a [[subharmonic function|Z1MHMGFNFW]] on $\\C$ that is bounded above is constant.\nThis stronger form fails in $\\R^n$ for $n \\ge 3$, where $-\\min(1, \\abs{x}^{2-n})$ is subharmonic, bounded above, and not constant.\n\n## Explanations\n\n### L1 human\n\nA harmonic function on the whole plane cannot stay below a ceiling unless it is flat.\nThink of a membrane stretched over the entire plane: if it never rises above some height, it must be level everywhere.\n\n### L2 human\n\nThe proof shows a typical planar technique: pass to a holomorphic function with real part $u$ and use complex analysis.\nThe theorem gives quick uniqueness results, for example for harmonic functions on $\\C$ with a given one-sided bound.\nIts subharmonic form says that $\\C$ is a parabolic surface in potential theory: it carries no nonconstant negative subharmonic function.\n",
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  "name": "Liouville's theorem for planar harmonic functions",
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