A TWO-PIECE PROPERTY FOR FREE BOUNDARY MINIMAL SURFACES IN THE BALL

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 Gruter and Simon. Our first result gives evidence to a conjecture by Fraser and Li.