Compactness of the space of embedded minimal surfaces with free boundary   in three-manifolds with nonnegative Ricci curvature and convex boundary
  We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies apriori area and curvature estimates for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness result for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed. 
Ailana Fraser and Martin Li
math.DG