Symmetries and regularity for holomorphic maps between balls

Abstract

Let $f:{\mathbb{B}}^n\to {\mathbb{B}}^N$ be a holomorphic map. We study subgroups ${\Gamma }_f\subseteq \mathrm{Aut}({\mathbb{B}}^n)$ and $T_f\subseteq \mathrm{Aut}({\mathbb{B}}^N)$. When $f$ is proper, we show both these groups are Lie subgroups. When ${\Gamma }_f$ contains the center of $\mathbf{U}(n)$, we show that $f$ is spherically equivalent to a polynomial. When $f$ is minimal we show that there is a homomorphism $\Phi :{\Gamma }_f\to T_f$ such that $f$ is equivariant with respect to $\Phi$. To do so, we characterize minimality via the triviality of a third group $H_f$. We relate properties of $\mathrm{Ker}(\Phi )$ to older results on invariant proper maps between balls. When $f$ is proper but completely non-rational, we show that either both ${\Gamma }_f$ and $T_f$ are finite or both are noncompact.