Uniqueness of a Complete Minimal Surfaces

Question

Let $R$ be a complete Riemannian Manifold of dimension $n$, let $S \subseteq R$ be homeomorphic to the sphere of dimension $k < n$, and let $M$ be a minimal surface of $R$ which is homeomorphic to the ball of dimension $k + 1$ and has $S$ as its boundary. If we allow self-intersections, does this minimal surface extend to some unique complete minimal surface? If $k = 0$, then $S$ is a pair of points and our $M$ is a geodesic connecting these two points, and this geodesic can be extended past either point in a unique way by moving "straight" past the points, and doing this process indefinitely leads to a unique geodesic which is either topologically a line or a circle (possibly with self-intersections), both of which are complete. I was wondering if this idea extends for higher dimensions. Perhaps the requirements about the boundary being homeomorphic to a $k$-sphere and the minimal surface being homeomorphic to a $k+1$-ball could be laxed.