Conformal extensions of functions defined on arbitrary subsets of Riemann surfaces

TL;DR

This paper establishes conditions under which functions defined on arbitrary subsets of Riemann surfaces can be extended conformally, clarifying the equivalence of definitions of analytic arcs in the complex plane.

Contribution

It provides new criteria for conformal extension of functions on arbitrary subsets of Riemann surfaces, linking different notions of analytic arcs.

Findings

Conditions for conformal extension are identified.

Equivalence of two common definitions of analytic arcs in ${ m extbf{C}}$ is proven.

The results facilitate understanding of analytic continuation on Riemann surfaces.

Abstract

For a function defined on an arbitrary subset of a Riemann surface, we give conditions which allow the function to be extended conformally. One folkloric consequence is that two common definitions of an analytic arc in ${\mathbb C}$ are equivalent. Key-words: analytic continuation; analytic arc.