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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $B \\subset \\C^n$ be the [[unit ball|PDQ9KW9FAP]].\nEvery [[proper holomorphic map|TM528Q96ZD]] $F \\colon B \\to B$ is a [[biholomorphism|FX92PP3HZR]] of $B$ onto itself.\n\n## Remarks\n\nA proper holomorphic self-map of $B$ is a finite branched covering.\nOne proof shows that it has no branching: near a suitable boundary point $F$ is a local biholomorphism that maps a piece of the sphere $\\partial B$ into $\\partial B$.\nSuch a local map is the restriction of an automorphism of $B$, a rigidity property of the sphere that goes back to Poincaré.\nBy the identity theorem, $F$ itself is that automorphism.\n\nThe hypothesis $n \\ge 2$ is necessary.\nFor $n = 1$ the map $z \\mapsto z^2$ is a proper holomorphic self-map of the unit disc that is not injective.\n\nThe shape of the ball matters too.\nThe map $(z_1, z_2) \\mapsto (z_1, z_2^2)$ is a proper holomorphic self-map of the unit polydisc in $\\C^2$ that is not injective.\n\nThe target dimension matters as well: the map $(z_1, z_2) \\mapsto (z_1^2, \\sqrt{2}\\, z_1 z_2, z_2^2)$ sends the unit ball of $\\C^2$ properly into the unit ball of $\\C^3$.\nCheck: $\\abs{z_1}^4 + 2\\abs{z_1 z_2}^2 + \\abs{z_2}^4 = (\\abs{z_1}^2 + \\abs{z_2}^2)^2$, which tends to $1$ exactly when $\\abs{z_1}^2 + \\abs{z_2}^2$ does.\n\n## Explanations\n\n### L1 human\n\nIn one variable the unit disc has many maps onto itself that wrap it several times, such as $z \\mapsto z^2$.\nAlexander's theorem says that the ball in two or more complex variables has none.\nAny holomorphic map from the ball to itself that sends the boundary to the boundary is a symmetry of the ball, one to one and onto.\n\n### L2 human\n\nThe theorem is a basic example of rigidity in several complex variables.\nCombined with the description of the automorphisms of $B$, it lists all proper self-maps of the ball.\nThey are the maps $z \\mapsto U \\varphi_a(z)$, with $U$ unitary and $\\varphi_a$ the automorphism exchanging $a$ and $0$.\nIt started the study of proper holomorphic maps between balls of different dimensions, where, as the example above shows, many non-trivial maps exist.\n",
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