Some Properties of the Intersection of Free Boundary Minimal Hypersurfaces in Euclidean Balls

TL;DR

This paper proves that any two free boundary minimal hypersurfaces in a Euclidean ball must intersect within any half-ball, extending the Frankel property and establishing a two-piece property for such hypersurfaces.

Contribution

It establishes a strong intersection property for free boundary minimal hypersurfaces and demonstrates the two-piece property in the Euclidean ball, extending previous Frankel-type results.

Findings

Any two free boundary minimal hypersurfaces intersect in any half-ball.

Equatorial disks divide minimal hypersurfaces into two connected parts.

Extension of the Frankel property to free boundary minimal hypersurfaces.

Abstract

In this work, we prove that any two free boundary minimal hypersurfaces in the unit Euclidean ball have an intersection point in any half-ball. This is a strong version of the Frankel property proved by A. Fraser and M. Li \cite{FRLI}. As a consequence, we obtain the two-piece property for free boundary minimal hypersurfaces in the unit ball: every equatorial disk divides any compact minimal hypersurface with free boundary in the unit ball in two connected pieces.