A two-piece property for free boundary minimal hypersurfaces in the $(n+1)$-dimensional ball

TL;DR

This paper proves a two-piece property for free boundary minimal hypersurfaces in Euclidean balls, showing they are divided into exactly two parts by hyperplanes through the origin, and characterizes certain mean convex regions.

Contribution

It establishes a two-piece property for free boundary minimal hypersurfaces and characterizes mean convex regions containing equatorial disks, supporting a conjecture by Fraser and Li.

Findings

Hyperplanes through the origin divide such hypersurfaces into two parts.

Regions with mean convex boundary containing an equatorial disk are half-balls.

Supports Fraser and Li's conjecture in any dimension.

Abstract

We prove that every hyperplane passing through the origin in $\rr^{n+1}$ divides an embedded compact free boundary minimal hypersurface of the euclidean $(n+1)$-ball in exactly two connected hypersurfaces. We also show that if a region in the $(n+1)$-ball has mean convex boundary and contains a nullhomologous $(n-1)$-dimensional equatorial disk, then this region is a closed halfball. Our first result gives evidence to a conjecture by Fraser and Li in any dimension.