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

Vanderson Lima, Ana Menezes

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.

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 divides an embedded compact free boundary minimal hypersurface of the euclidean -ball in exactly two connected hypersurfaces. We also show that if a region in the -ball has mean convex boundary and contains a nullhomologous -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.

Paper Structure

This paper contains 6 sections, 7 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Free boundary minimal hypersurfaces
  4. Integer rectifiable varifolds
  5. Minimizing Currents with Partially Free Boundary
  6. The two-piece property and other results

Key Result

Lemma 1

Let $\Sigma$ be a partially free boundary minimal hypersurface in $\mathop{\mathrm{B}}\nolimits^{n+1}$ of finite area and such that the singular set $\mathcal{S}_\Sigma=\overline{\Sigma}\setminus \Sigma$ satisfies $\mathcal{S}_{\Sigma}=\mathcal{S}_0\cup\mathcal{S}_1$, where $\mathcal{S}_0 \subset\ov

Figures (2)

  • Figure 1: $\Omega=\Omega_1\cup\Omega_2$.
  • Figure 2: In this region $W$, any $(n-1)-$dimensional equatorial disk $\Upsilon\subset W$ is nullhomologous.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Claim 1
  • Lemma 2
  • Proposition 2
  • Theorem 1
  • ...and 7 more