2024/01/12 by Abdullah Al Helal, Helal, Abdullah Al, Jiř́í Lebl +3 · 1 citation
Mathematics · #32H02 (Secondary) #32H35 (Primary) 32A08 #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematics and Applications #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2401.06364
openalex publication_date 2024/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider proper holomorphic maps of ball complements and differences in complex euclidean spaces of dimension at least two. Such maps are always rational, which naturally leads to a related problem of classifying rational maps taking concentric spheres to concentric spheres, what we call m-fold sphere maps; a proper map of the difference of concentric balls is a 2-fold sphere map. We prove that proper maps of ball complements are in one to one correspondence with polynomial proper maps of balls taking infinity to infinity. We show that rational m-fold sphere maps of degree less than m (or polynomial maps of degree m or less) must take all concentric spheres to concentric spheres and we provide a complete classification of them. We prove that these degree bounds are sharp.