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.