On boundary extension of mappings of Riemannian surfaces in terms of prime ends

TL;DR

This paper studies non-homeomorphic Sobolev class mappings of Riemannian surfaces, providing distortion estimates and conditions for continuous boundary extension via prime ends.

Contribution

It introduces new conditions under which such mappings can be extended continuously to the boundary using prime ends.

Findings

Derived distortion estimates for moduli of curve families.

Proved boundary extension results for Sobolev mappings.

Established conditions for continuous extension to boundary via prime ends.

Abstract

We investigate non-homeomorphic mappings of Riemannian surfaces of Sobolev class. We have obtained some estimates of distortion of moduli of families of curves. We have proved that, under some conditions, these mappings have a continuous extension to a boundary of a domain in terms of prime ends.