Quasiconformal curves and quasiconformal maps in metric spaces

Abstract

In this paper we study quasiconformal curves which are a special case of quasiregular curves. Namely embeddings $\Omega \to {\mathbb{R}}^m$ from some domain $\Omega \subset {\mathbb{R}}^n$ to ${\mathbb{R}}^m$, where $n\le m$, which belong in a suitable Sobolev class and satisfy a certain distortion inequality for some smooth, closed and non-vanishing $n$-form in ${\mathbb{R}}^m$. These mappings can be seen as quasiconformal mappings between $\Omega$ and $f(\Omega )$. We prove that a quasiconformal curve always satisfies the analytic definition of quasiconformal mappings and the lower half of the modulus inequality. Moreover, we give a sufficient condition for a quasiconformal curve to satisfy the metric definition of quasiconformal mappings. We also show that a quasiconformal map from $\Omega$ to $f(\Omega )\subset {\mathbb{R}}^m$ is a quasiconformal $\omega$ curve for some form $\omega$ under suitable assumptions. Finally, we show the same is true when we equip the target space $f(\Omega )$ with its intrinsic metric instead of the Euclidean one.