A Jordan Curve Spanned by a Complete Minimal Surface

Abstract

In this paper we construct complete (conformal) minimal immersions $f: {\mathbb D} \longrightarrow {\mathbb R}^3$ which admit continuous extensions to the closed disk, $F: \overline{\mathbb D} \longrightarrow {\mathbb R}^3$. Moreover, $F_{|{\mathbb S}^1}: {\mathbb S}^1 \rightarrow F({\mathbb S}^1)$ is a homeomorphism and $F({\mathbb S}^1)$ is a (non-rectifiable) Jordan curve with Hausdorff dimension 1.

It turns out that the set of Jordan curves $F({\mathbb S}^1)$ constructed by the above procedure is dense in the space of Jordan curves of $\mathbb{R}^3$ with the Hausdorff metric.