Quasicircles and Dirichlet finite harmonic functions on Riemann surfaces

Abstract

A Jordan curve in $\mathbb{C}$ is called a quasicircle if it is an image of the unit circle by a quasiconformal self-mapping of $\mathbb{C}$. Quasicircles are characterized by an extendability condition on Dirichlet finite harmonic functions. In this chapter, we improve and generalize the condition for Jordan curves to be quasicircles on Riemann surfaces. Also, our results are generalizations of a theorem by Schippers and Staubach (2020), which shows the extendability property for a Jordan curve on a compact Riemann surface.