تقرير
Proper maps of ball complements & differences and rational sphere maps
| العنوان: | Proper maps of ball complements & differences and rational sphere maps |
|---|---|
| المؤلفون: | Helal, Abdullah Al, Lebl, Jiří, Nandi, Achinta Kumar |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Complex Variables, 32H35 (Primary) 32A08, 32H02 (Secondary) |
| الوصف: | We consider proper holomorphic maps of complements and differences of balls in complex euclidean space of dimension at least $2$. We prove that every proper map of ball complements gives a polynomial proper map of balls, and conversely, every polynomial proper map of balls whose norm goes to infinity at infinity is a proper map of ball complements. The case of ball differences is more complex and naturally leads to what we call rational $m$-fold sphere maps, that is, rational maps taking $m$ zero-centric spheres to $m$ zero-centric spheres. A proper map of the difference of zero-centric balls is automatically a rational $2$-fold sphere map. We show that a polynomial $m$-fold sphere map of degree $m$ or less is an $\infty$-fold sphere map, that is, a map that takes infinitely many (and hence every) zero-centric spheres to zero-centric spheres. Similarly, every rational $m$-fold sphere map of degree less than $m$ is an $\infty$-fold sphere map. We then show that $\infty$-fold sphere maps are up to a unitary transformation direct sums of a finite number of homogeneous sphere maps. We construct rational $m$-fold degree-$m$ sphere maps that do not take any other zero-centric sphere to a zero-centric sphere. In particular, every first-degree rational proper map of the difference of zero-centric balls is a unitary composed with an affine linear embedding; however, there exist nonpolynomial second-degree rational maps of a difference of balls. Comment: 21 pages, Theorem 1.1 is corrected, extraneous examples of Section 5 are removed |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2401.06364 |
| رقم الأكسشن: | edsarx.2401.06364 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |