  In this paper we find approximate solutions of certain Riemann-Hilbert
boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null
holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand
we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$
which are normalized by any given bordered Riemann surface and have Jordan
boundaries. We also furnish complete conformal proper minimal immersions from
any given bordered Riemann surface to any smoothly bounded, strictly convex
domain of $\mathbb{R}^n$ which extend continuously up to the boundary; for
$n\ge 5$ we find embeddings with these properties.

Differential Geometry