Geodesic
The exterior Plateau problem in higher codimension
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 44-56

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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. 
@article{CMFD_2006_17_a3,
     author = {F. Tomi and L. P. Jorge},
     title = {The exterior {Plateau} problem in higher codimension},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {44--56},
     publisher = {mathdoc},
     volume = {17},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2006_17_a3/}
}
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F. Tomi; L. P. Jorge. The exterior Plateau problem in higher codimension. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 44-56. http://geodesic.mathdoc.fr/item/CMFD_2006_17_a3/