INVARIANT PROPER HOLOMORPHIC MAPS BETWEEN BALLS

In this paper we consider proper holomorphic maps between balls that are invariant under the action of finite fixed point free unitary groups. Forstneric showed that given any such group (representation) there exists such a map. He showed that if we also require the map to be smooth to the boundary, then many available groups are ruled out. In this paper we prove the following theorem: If f is a proper holomorphic map between balls that is smooth to the boundary and invariant under the action of such a group, then necessarily that group is cyclic. We rule out some of these cases as well. We discuss some related theory concerning group-invariant proper holomorphic maps between balls. One result uses a modification of a technique due to Low, that embeds smooth strongly pseudoconvex domains properly into balls, to produce new examples of group-invariant maps.