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

## TL;DR

This paper establishes a two-piece property for free boundary minimal surfaces in the 3-ball, proves a halfball characterization under certain conditions, and addresses regularity at corners of currents in free boundary problems.

## Contribution

It introduces a new two-piece property for free boundary minimal surfaces and proves a halfball characterization, providing evidence for a conjecture by Fraser and Li.

## Key findings

- Every plane through the origin divides the surface into two parts.
- Regions with mean convex boundary containing a nullhomologous diameter are halfballs.
- Regularity at corners of currents in free boundary problems is established.

## 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.