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

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