The exterior Plateau problem in higher codimension

We prove existence theorems for two-dimensional noncompact complete minimal surfaces in $\mathbb R^n$ of annular type, which span a given contour and have a finite total curvature end and prescribed asymptotical behavior. For arbitrary rectifiable Jordan curves, we show the existence of such surfaces with a flat end, i.e., within bounded distance from a 2-plane. For more restricted classes of curves, we prove the existence of minimal surfaces with higher multiplicity flat ends as well as of surfaces with polynomial-type nonflat ends.