Curvature estimates for minimal surfaces with total boundary curvature less than 4$\pi $

Abstract

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4φ. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also prove that the set of simple closed curves with total curvature less than 4φ and which do not bound an orientable compact embedded minimal surface of genus greater than g, for any given g,isopenin the C 2'α topology. © 2009 American Mathematical Society.