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  "body": "\n## Statement\n\nLet $D \\subset \\C^n$ be a bounded domain, that is, a bounded connected open set.\nLet $\\Aut(D)$ be the group of [[biholomorphisms|FX92PP3HZR]] $D \\to D$, with the topology of uniform convergence on compact subsets.\nThen $\\Aut(D)$ is a real [[Lie group|7JBRGCVJRB]], and it acts properly on $D$.\nIn particular, for each $p \\in D$ the isotropy group $\\set{f \\in \\Aut(D) : f(p) = p}$ is compact.\n\n## Remarks\n\nThe theorem is due to Henri Cartan.\nSince $D$ is bounded, [[Montel's theorem|ZVFVBVJMCB]] makes families of automorphisms normal, and a limit argument shows that the action is proper.\nBy [[Cartan's uniqueness theorem|EQHPSKC5SY]], an automorphism $f$ with $f(p) = p$ is determined by its derivative $f'(p)$.\nSo the isotropy group embeds as a compact subgroup of $GL_n(\\C)$, and the whole group is locally compact and finite-dimensional.\n\nExample: for the unit disc $\\Delta$, $\\Aut(\\Delta)$ consists of the maps $z \\mapsto e^{i\\theta}(z - a)/(1 - \\bar a z)$ with $\\theta \\in \\R$ and $\\abs{a} \u003c 1$.\nIt is a $3$-dimensional Lie group, isomorphic to $PSU(1, 1)$, and the isotropy group of $0$ is the circle of rotations.\n\nBoundedness cannot be dropped.\nFor $n \\ge 2$, $\\Aut(\\C^n)$ contains the maps $(z_1, z_2, \\dots) \\mapsto (z_1, z_2 + g(z_1), \\dots)$ for every entire function $g$, so it is not a finite-dimensional Lie group.\n\n## Explanations\n\n### L1 human\n\nThe holomorphic symmetries of a bounded region in $\\C^n$ form a group.\nThe theorem says this group is as nice as a group of matrices: it is a smooth finite-dimensional manifold, and its operations are smooth.\nMoreover, the symmetries fixing a point form a compact group.\n\n### L2 human\n\nThis is the starting point for the study of homogeneous and symmetric bounded domains.\nIf $\\Aut(D)$ acts transitively, then $D \\cong \\Aut(D)/K$ with $K$ compact, and $D$ becomes a homogeneous space of a Lie group.\nCombined with the invariant Bergman metric, this leads to Cartan's classification of bounded symmetric domains.\nFor unbounded domains the group can be infinite-dimensional, as for $\\C^n$.\n",
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