A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

Abstract

In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ∂ M . These minimal surfaces are either disjoint from ∂ M or meet ∂ M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ∂ M . We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, 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 . Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.