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Proper maps of ball complements & differences and rational sphere maps

Abdullah Al Helal, Jiří Lebl, Achinta Kumar Nandi

TL;DR

The paper shows that proper holomorphic maps between ball complements and between ball differences in $\mathbb{C}^n$ (with $n\ge 2$) are necessarily rational and can be effectively analyzed via polynomial and rational $m$-fold sphere maps. It establishes a precise correspondence between ball-complement maps and polynomial ball maps sending infinity to infinity, and provides a complete, sharp classification of rational $m$-fold sphere maps, including both polynomial and nonpolynomial examples and a unitary-decomposition structure for $\infty$-fold maps. Using Newton-polynomial representations, the authors derive concrete degree constraints and show that low-degree maps must preserve concentric spheres, while also constructing nontrivial $k$-fold sphere maps that are not $(k+1)$-fold. The results yield a detailed understanding of the existence, structure, and limitations of such maps, including nonexistence results for maps between ball differences and ball complements in higher dimensions.

Abstract

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.

Proper maps of ball complements & differences and rational sphere maps

TL;DR

The paper shows that proper holomorphic maps between ball complements and between ball differences in (with ) are necessarily rational and can be effectively analyzed via polynomial and rational -fold sphere maps. It establishes a precise correspondence between ball-complement maps and polynomial ball maps sending infinity to infinity, and provides a complete, sharp classification of rational -fold sphere maps, including both polynomial and nonpolynomial examples and a unitary-decomposition structure for -fold maps. Using Newton-polynomial representations, the authors derive concrete degree constraints and show that low-degree maps must preserve concentric spheres, while also constructing nontrivial -fold sphere maps that are not -fold. The results yield a detailed understanding of the existence, structure, and limitations of such maps, including nonexistence results for maps between ball differences and ball complements in higher dimensions.

Abstract

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 -fold sphere maps; a proper map of the difference of concentric balls is a -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 -fold sphere maps of degree less than (or polynomial maps of degree 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.
Paper Structure (6 sections, 23 theorems, 66 equations)

This paper contains 6 sections, 23 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Proper mappings of ball complements
  3. m-fold sphere maps
  4. Polynomial m-fold sphere maps
  5. Rational m-fold sphere maps
  6. Proper mappings of ball differences

Key Result

Theorem 1.1

Suppose $f \colon \mathbb{C}^n \setminus \overline{\mathbb B_n} \to \mathbb{C}^N \setminus \overline{\mathbb B_N}$, $n \geq 2$, is a proper holomorphic map. Then $f$ is a polynomial map, and when this polynomial is restricted to $\mathbb B_n$, it gives a proper map to $\mathbb B_N$. Conversely, supp

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 35 more