New Brief Proofs of the Gulliver–Osserman–Royden Theorem

Abstract

We would like to present very much simplified proofs of versions of the Gulliver–Osserman–Royden (GOR) theorem (1973), in the case Γ is C ^2,α smooth. In the first proof instead of employing a topological theory of ramified coverings used in (GOR), we introduce a new analytical method of root curves. The surprising aspect of this proof is that it connects the issue of the existence of analytical false interior branch points with boundary branch points. We should note that this fact was also observed by F. Tomi (to appear) who has found his own very brief proof of (GOR) in the case Γ∈C ^2,α which we also include.