  Let N be a complete, homogeneously regular Riemannian manifold of dimension
greater than 2 and let M be a compact submanifold of N. Let $\Sigma$ be a
compact orientable surface with boundary. We show that for any continuous $f:
(\Sigma, \partial \Sigma) \rightarrow (N, M)$ for which the induced
homomorphism on certain fundamental groups is injective, there exists a
branched minimal immersion of $\Sigma$ solving the free boundary problem
$(\Sigma, \partial \Sigma) \rightarrow (N, M)$, and minimizing area among all
maps which induce the same action on the fundamental groups as f. Furthermore,
under certain nonnegativity assumptions on the curvature of a 3-manifold N and
convexity assumptions on M which is the boundary of N, we derive bounds on the
genus, number of boundary components and area of any compact two-sided minimal
surface solving the free boundary problem with low index.
