Minimal immersions of compact bordered Riemann surfaces with free   boundary
  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. 
Jingyi Chen, Ailana Fraser, Chao Pang
math.DG