Alexander's theorem on proper self-maps of the ball

Remarks

A proper holomorphic self-map of $B$ is a finite branched covering. One 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 $\partial B$. Such a local map is the restriction of an automorphism of $B$, a rigidity property of the sphere that goes back to Poincaré. By the identity theorem, $F$ itself is that automorphism.

The hypothesis $n \ge 2$ is necessary. For $n = 1$ the map $z \mapsto z^2$ is a proper holomorphic self-map of the unit disc that is not injective.

The shape of the ball matters too. The 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.

The 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 $\C^3$. Check: $\abs{z_1}^4 + 2\abs{z_1 z_2}^2 + \abs{z_2}^4 = (\abs{z_1}^2 + \abs{z_2}^2)^2$, which tends to $1$ exactly when $\abs{z_1}^2 + \abs{z_2}^2$ does.

Going deeper

For a student

The theorem is a basic example of rigidity in several complex variables. Combined with the description of the automorphisms of $B$, it lists all proper self-maps of the ball. They are the maps $z \mapsto U \varphi_a(z)$, with $U$ unitary and $\varphi_a$ the automorphism exchanging $a$ and $0$. It started the study of proper holomorphic maps between balls of different dimensions, where, as the example above shows, many non-trivial maps exist.