On proper holomorphic mappings from domains with $\bf T$-action

Abstract

We describe the branch locus of a proper holomorphic mapping between two smoothly bounded pseudoconvex domains of finite type in $\mathbb{C}^2$ under the assumption that the first domain admits a transversal holomorphic action of the unit circle. As an application we show that any proper holomorphic self-mapping of a smoothly bounded pseudoconvex complete circular domain of finite type in $\mathbb{C}^2$ is biholomorphic.