Mathematical Research Letters

Volume 25 (2018)

Number 5

Symmetries and regularity for holomorphic maps between balls

Pages: 1389 – 1404



John P. D’Angelo (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)

Ming Xiao (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.; and Department of Mathematics, University of California at San Diego)


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 $\mathrm{U}(n)$, 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.

The first author acknowledges support from NSF Grant DMS 13-61001.

Received 17 April 2017

Accepted 15 January 2018

Published 1 February 2019