A two-piece property for free boundary minimal surfaces in the ball

Abstract

We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean $3$-ball in exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains a nullhomologous diameter, then this region is a closed halfball. Moreover, we prove the regularity at the corners of currents minimizing a partially free boundary problem by following ideas by Gr\"uter and Simon. Our first result gives evidence to a conjecture by Fraser and Li.