Proper maps of annuli
Abdullah Al Helal, Jiri Lebl, Achinta Kumar Nandi
TL;DR
This work investigates proper holomorphic maps between annuli $\mathbb{A}_{n,r}$ and $\mathbb{A}_{N,R}$ in $\mathbb{C}^n$ with automorphism group $U(n)$. Leveraging Hartogs extension and Forstneric's rationality results, the maps are shown to be rational and extend to proper ball maps, enabling a correspondence with rational maps that take spheres to spheres. A sharp first gap is established: for $2 \le n < N < \binom{n+1}{2}$, such maps must be unitarily equivalent to an affine embedding, with the homogeneous map $H_2$ achieving $N=\binom{n+1}{2}$. The authors introduce the general hyperplane rank $k_f$ and prove that $k_f = N-1$ if and only if the map is homogeneous $f = U H_d$ with $N=\binom{n+d-1}{d}$, leading to a complete analogue of Faran’s result for the $(2,3)$ case. The paper thus provides a framework for classifying annulus maps, including a precise dichotomy between affine embeddings and homogeneous maps, and outlines open questions on degree bounds and non-homogeneous constructions.
Abstract
We study rational 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 $N-1$. This result is really a result classifying homogeneous proper maps of balls. A homogeneous proper ball map takes 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.