Abstract. Let N be a complete, homogeneously regular Riemannian manifold of dimN ≥ 3 and let M be a compact submanifold of N . Let Σ be a compact orientable surface with boundary. We show that for any continuous f : (Σ, ∂Σ) → (N, M ) for which the induced homomorphism f * on certain fundamental groups is injective, there exists a branched minimal immersion of Σ solving the free boundary problem (Σ, ∂Σ) → (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 the boundary M = ∂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.

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