Proper maps of annuli

We study proper holomorphic maps of annuli in complex Euclidean spaces, that is, domains with $U(n)$ as the automorphism group. By the Hartogs phenomenon and a result of Forstnerič, such maps are always rational and extend to proper maps of balls. We first prove that a proper map of annuli from $n$ dimensions to $N$ dimensions where $N < \binom{n+1}{2}$ is always an affine embedding. This inequality is sharp as the homogeneous map of degree 2 satisfies $N=\binom{n+1}{2}$. Next we find a necessary and sufficient condition for a map to be homogeneous: A proper map of annuli is homogeneous if and only if its general hyperplane rank, the affine dimension of the image of a general hyperplane, is exactly $N-1$. As a corollary, we obtain a classification of homogeneous proper maps of balls. A homogeneous proper ball map takes all spheres centered at the origin to spheres centered at the origin. We show that if a proper ball map has general hyperplane rank $N-1$ and takes one sphere centered at the origin to a sphere centered at the origin, then it is homogeneous. Another corollary of this result is a complete classification of proper maps of annuli from dimension 2 to dimension 3. Finally, we give a complete normal form of rational proper maps of annuli of degree 2.

💡 Research Summary

The paper investigates proper holomorphic maps between annuli in complex Euclidean spaces, i.e. domains of the form
(A_{n,r}={z\in\mathbb C^{n}: r<|z|<1}) and (A_{N,R}). For (n>1) the automorphism group of an annulus is the unitary group (U(n)); this high symmetry makes the problem both tractable and distinct from the classical ball case.

Rationality and Extension.
Using the Hartogs phenomenon, any proper map (f:A_{n,r}\to A_{N,R}) extends holomorphically across the inner ball (rB_{n}) and therefore to a map on the whole unit ball (B_{n}). By a theorem of Forstnerič, such extensions are always rational, with a degree bounded by a function (D(n,N)). Moreover, the map sends the inner sphere (rS^{2n-1}) to the inner sphere (RS^{2N-1}) and the outer sphere to the outer sphere, so the restriction to the annulus is completely characterized by a rational sphere map.

Gap Theorem (Theorem 1.1).
If (2\le n<N<\binom{n+1}{2}) then every proper map is an affine embedding. Explicitly, \