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

TL;DR

This paper demonstrates that in high-dimensional circular cones with sufficiently wide opening angles, certain axial hyperplanes are area-minimizing with free-boundary conditions, challenging previous theorems.

Contribution

It introduces a calibration-based method to prove minimality of free-boundary hyperplanes in high-dimensional cones, providing counterexamples to a recent Vertex-skipping Theorem.

Findings

Existence of a threshold angle 5(n) for minimality in high dimensions

Construction of free-boundary minimal hyperplanes in wide cones

Counterexamples to the Vertex-skipping Theorem for na0a4 4

Abstract

Consider an $(n+1)$-dimensional circular cone with opening angle $\alpha \in (0,\pi)$. Using a free-boundary adaptation of the classical calibration method, we prove that, for $n \geq 4$, there exists a threshold $\bar{\alpha}(n) \in (0,\pi)$ such that if $\alpha \geq \bar{\alpha}(n)$, that is, the cone is wide enough, the intersection of the cone with an axial hyperplane is area-minimizing with respect to free-boundary variations inside the cone. This provides a counterexample to a recent Vertex-skipping Theorem proved by the author in collaboration with G.P. Leonardi, at least for $n\geq4$.