Proper maps of annuli

TL;DR

This paper characterizes proper holomorphic maps of annuli in complex Euclidean spaces, establishing conditions for their homogeneity, classification, and normal forms, with implications for maps between balls and annuli.

Contribution

It provides a sharp inequality for proper maps of annuli, characterizes homogeneity via hyperplane rank, and classifies proper maps in specific dimensions and degrees.

Findings

Proper maps of annuli with N < binom(n+1,2) are affine embeddings.

Homogeneity is characterized by hyperplane rank being N-1.

Complete classification of proper maps from 2D to 3D annuli and degree 2 rational maps.

Abstract

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 Forstneri\v{c}, 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.