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

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.

Final version. Accepted for publication in Communications in Analysis and Geometry

References in corpus (3)

• Existence of infinitely many free boundary minimal hypersurfaces
• Multiplicity one for min-max theory in compact manifolds with boundary and its applications
• Some Properties of the Intersection of Free Boundary Minimal Hypersurfaces in Euclidean Balls