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  "body": "\n## Statement\n\nLet $X$ be a simply connected [[Riemann surface|5K0N8QMV6Q]], that is, its [[fundamental group|S1XK3HXHFX]] is trivial.\nThen $X$ is [[biholomorphic|8AJEMCZ29B]] to exactly one of the following three Riemann surfaces:\n\n1. the [[Riemann sphere|0CDMXGPSES]] $\\hat\\C$;\n2. the [[complex plane|0CTE7A31XQ]] $\\C$;\n3. the open [[unit disc|8G5ABXRK85]] $\\mathbb{D}$.\n\n## Remarks\n\nThe three surfaces are pairwise not biholomorphic: $\\hat\\C$ is compact and the others are not, and a holomorphic map $\\C \\to \\mathbb{D}$ is constant by Liouville's theorem.\nApplied to the [[universal cover|Y1YPE6PM52]], the theorem shows that every Riemann surface $X$ is biholomorphic to a quotient $\\tilde X / \\Gamma$.\nHere $\\tilde X$ is one of the three surfaces and $\\Gamma$ is a group of automorphisms of $\\tilde X$ acting freely and properly discontinuously.\nOnly $\\hat\\C$ itself is covered by $\\hat\\C$, and only $\\C$, $\\C \\setminus \\set{0}$ and the tori $\\C / \\Lambda$ are covered by $\\C$.\nEvery other Riemann surface is a quotient $\\mathbb{D} / \\Gamma$ with $\\Gamma$ a Fuchsian group.\nFor simply connected open subsets of $\\C$ the theorem contains the [[Riemann mapping theorem|02WHMEG5ZR]].\n\nThe theorem was proved in 1907, independently by Koebe and by Poincaré.\nOne proof constructs a harmonic Green's function or a harmonic dipole function on $X$ and uses it to build a holomorphic injection into $\\hat\\C$.\n\nExample: $\\C \\setminus \\set{0}$ is not simply connected; its universal cover is $\\exp \\colon \\C \\to \\C \\setminus \\set{0}$, and $\\C \\setminus \\set{0} \\cong \\C / 2\\pi i \\Z$.\n\n## Explanations\n\n### L1 human\n\nThere are infinitely many Riemann surfaces, but only three that have no loops that cannot be shrunk: the sphere, the plane and the disc.\nEvery other Riemann surface is obtained from one of these three by rolling it up, that is, by identifying points that differ by a group of symmetries.\nSo the study of all Riemann surfaces reduces to the study of these three and their symmetry groups.\n\n### L2 human\n\nThe theorem gives every Riemann surface a metric of constant curvature $+1$, $0$ or $-1$ compatible with its complex structure.\nCompact surfaces of genus $0$ are spherical, genus $1$ flat, and genus at least $2$ hyperbolic, a fact that links complex analysis to hyperbolic geometry.\nIt is the starting point of Teichmüller theory, because a complex structure on a surface of genus $g \\ge 2$ becomes a Fuchsian group up to conjugation.\n",
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