Symmetries and regularity for holomorphic maps between balls

TL;DR

This paper investigates the symmetry groups of holomorphic maps between unit balls, revealing their structure, conditions for polynomial equivalence, and the relationship between properness, minimality, and rationality.

Contribution

It characterizes symmetry groups of holomorphic maps between balls, establishes conditions for polynomial equivalence, and links properness and non-rationality to group properties.

Findings

Both symmetry groups are Lie subgroups when the map is proper.

Maps with certain symmetry contain the center of the unitary group and are polynomial.

Proper but non-rational maps have symmetry groups that are either both finite or both noncompact.

Abstract

Let $f:{\mathbb B}^n \to {\mathbb B}^N$ be a holomorphic map. We study subgroups $\Gamma_f \subseteq {\rm Aut}({\mathbb B}^n)$ and $T_f \subseteq {\rm Aut}({\mathbb B}^N)$. When $f$ is proper, we show both these groups are Lie subgroups. When $\Gamma_f$ 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 $\Phi:\Gamma_f \to T_f$ 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 ${\rm Ker}(\Phi)$ 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.