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Minimality of free-boundary axial hyperplanes in high dimensional circular cones via calibration

Giacomo Vianello

TL;DR

This work addresses the free-boundary minimality of axial hyperplanes in high-dimensional circular cones. It develops a calibration framework adapted to boundary-tangent conditions, constructing a divergence-free vector field $Z$ (via a two-step process that first yields a boundary calibration $Y$ and then extends by vertical projection) to compare the relative perimeters of a minimizer $E$ with competing sets. The main result shows that for $n\ge 4$ and $0<\lambda\le\bar{\lambda}(n)$, the axial half-space $E=\{(x,t): x_1>0\}$ minimizes the relative perimeter in the cone $\Omega_{\lambda}$, providing a higher-dimensional counterexample to Vertex-skipping type results. These findings illuminate the boundary regularity and stability aspects of free-boundary minimal surfaces in cones and highlight a dimension-dependent transition in behavior as $n$ increases.

Abstract

Consider an $(n+1)$-dimensional circular cone. Using a calibration argument, we prove that if $n \geq 4$ and the aperture of the cone is sufficiently large, the intersection of the cone with an axial hyperplane is area-minimizing with respect to free-boundary variations inside the cone.

Minimality of free-boundary axial hyperplanes in high dimensional circular cones via calibration

TL;DR

This work addresses the free-boundary minimality of axial hyperplanes in high-dimensional circular cones. It develops a calibration framework adapted to boundary-tangent conditions, constructing a divergence-free vector field (via a two-step process that first yields a boundary calibration and then extends by vertical projection) to compare the relative perimeters of a minimizer with competing sets. The main result shows that for and , the axial half-space minimizes the relative perimeter in the cone , providing a higher-dimensional counterexample to Vertex-skipping type results. These findings illuminate the boundary regularity and stability aspects of free-boundary minimal surfaces in cones and highlight a dimension-dependent transition in behavior as increases.

Abstract

Consider an -dimensional circular cone. Using a calibration argument, we prove that if and the aperture of the cone is sufficiently large, the intersection of the cone with an axial hyperplane is area-minimizing with respect to free-boundary variations inside the cone.
Paper Structure (10 sections, 5 theorems, 88 equations)

This paper contains 10 sections, 5 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Preliminary notions and basic notation
  4. Manifolds and $k$-forms.
  5. Riemannian manifolds.
  6. The Hodge-$\star$ operator.
  7. Locally finite perimeter sets under set operations
  8. Further notation and preliminary computations
  9. Construction of the calibration
  10. Minimality of $E$

Key Result

Theorem 1.1

For $n \geq 4$, consider the threshold parameter and assume that $0 < \lambda \leq \bar{\lambda}(n)$. Let also Then $E$ is a minimizer of the relative perimeter in $\Omega_{\lambda}$, that is, given a locally-finite perimeter set $F \subset \mathbb{R}^{n+1}$ satisfying $E \Delta F \subset\!\subset B_R$, for some $R>0$, one has

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • proof : Proof (of Theorem \ref{['thm:excal']}).
  • proof : Proof (of Theorem \ref{['thm:min']}).