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

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 $n\ge 4$, there exists a threshold $\bar{\alpha }(n)\in (0,\pi )$ such that if $\alpha \ge \bar{\alpha }(n)$, 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 $n\ge 4$.