Minimal immersions of compact bordered Riemann surfaces with free boundary

Abstract

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.