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

A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary