# Prime ends and mappings on Riemann surfaces

## TL;DR

This paper establishes criteria for when mappings with finite distortion between domains on Riemann surfaces can be extended continuously or homeomorphically to the boundary using prime ends.

## Contribution

It introduces new boundary extension criteria for finite distortion mappings on Riemann surfaces utilizing prime ends of Caratheodory.

## Key findings

- Criteria for continuous extension of mappings
- Criteria for homeomorphic extension
- Application of prime ends in boundary analysis

## Abstract

It is proved criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.