Let
N
N
be a complete, homogeneously regular Riemannian manifold of dim
N
≥
3
N \geq 3
and let
M
M
be a compact submanifold of
N
N
. Let
Σ
\Sigma
be a compact orientable surface with boundary. We show that for any continuous
f
:
(
Σ
,
∂
Σ
)
→
(
N
,
M
)
f: \left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right )
for which the induced homomorphism
f
∗
f_{*}
on certain fundamental groups is injective, there exists a branched minimal immersion of
Σ
\Sigma
solving the free boundary problem
(
Σ
,
∂
Σ
)
→
(
N
,
M
)
\left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right )
, and minimizing area among all maps which induce the same action on the fundamental groups as
f
f
. Furthermore, under certain nonnegativity assumptions on the curvature of a
3
3
-manifold
N
N
and convexity assumptions on the boundary
M
=
∂
N
M=\partial 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.