A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

In this paper, we develop a general existence theory for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold $M$ with boundary $\partial M$. The main feature of our result is that no convexity assumption is required on $\partial M$. Our proof uses a variant of the min-max construction first considered by Almgren and Pitts. Recently, Colding-De Lellis gave a simplified proof of the interior regularity and here, we prove the boundary regularity of the limiting embedded minimal surfaces at their free boundaries. In addition, we define a topological invariant, the filling genus, for compact 3-manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold $M$.