Proper maps and involutions of unit balls in Euclidean Levi-flat spaces

Abstract

As models of strictly pseudoconvex domains, we consider holomorphic functions on the unit ball $\ball{n}=\{z\in\C^n:|z|<1\}$. In particular, we focus on proper holomorphic maps $\ball{n}\to\ball{N}$. In the equidimensional case $N=n$, proper holomorphic maps are automorphisms. We discuss the parameters associated to automorphisms, and more generally involutions and their higher-order analogues.

We then define the mixed spaces $\ball{n,k}=\{(z,s)\in\C^n\times\R^k:|z|^2+|s|^2<1\}$, and address similar questions regarding proper maps, automorphisms, and involutions in the new setting. In particular, we show how to recover the parameters that determine an automorphism of $\ball{n,k}$ using the germ at $z=0$. We also specify necessary conditions on involutions in both the $\ball{n}$ and $\ball{n,k}$ settings.