Nonclassical minimizing surfaces with smooth boundary

Abstract

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $\Gamma \subset \mathbb{R}^4$ such that the unique area minimizing surface spanned by $\Gamma$ has infinite topology. Furthermore the metric is almost Kähler and the area minimizing surface is calibrated.