Year: 2017
Type: preprint
Abstract: Let $f:{\mathbb B}^n \to {\mathbb B}^N$ be a holomorphic map. We study subgroups $Γ_f \subseteq {\rm Aut}({\mathbb B}^n)$ and $T_f \subseteq {\rm Aut}({\mathbb B}^N)$. When $f$ is proper, we show both these groups are Lie subgroups. When $Γ_f$ contains the center of ${\bf U}(n)$, we show that $f$ is spherically equivalent to a polynomial. When $f$ is minimal we show that there is a homomorphism $Φ:Γ_f \to T_f$ such that $f$ is equivariant with respect to $Φ$. To do so, we characterize minimality via the triviality of a third group $H_f$. We relate properties of ${\rm Ker}(Φ)$ to older results on invariant proper maps between balls. When $f$ is proper but completely non-rational, we show that either both $Γ_f$ and $T_f$ are finite or both are noncompact.
Source: arXiv (Cornell University)
Authors: D'Angelo, John P., Xiao, Ming
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2770158223