On equicontinuity of families of mappings between Riemannian surfaces with respect to prime ends

Abstract

Given a domain of some Riemannian surface,we consider questions related to the possibility of a continuousextension to the boundary of one class of Sobolev mappings. It isproved that such maps have a continuous boundary extension in termsof prime ends, and under some additional restrictions their familiesare equicontinuous at inner and boundary points of the domain. Wehave separately considered the cases of homeomorphisms and mappingswith branching.