Abstract: We study rational holomorphic maps of annuli in complex euclidean spaces, which are domains with $U(n)$ as the automorphism group. 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 unitarily equivalent to an affine embedding. This inequality is sharp as the homogeneous map of degree 2 satisfies $N=\binom{n+1}{2}$. This yields the first gap interval for proper holomorphic maps between annuli, which surprisingly is much longer than the well-known first gap interval $(n, 2n - 1)$ for ball maps. This is a joint work with Prof.\ Jiří Lebl and Dr.\ Achinta Nandi.
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