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

Consider an $(n+1)$-dimensional circular cone with opening angle $α\in (0,π)$. Using a free-boundary adaptation of the classical calibration method, we prove that, for $n \geq 4$, there exists a threshold $\barα(n) \in (0,π)$ such that if $α\geq \barα(n)$, 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 $n\geq4$.

💡 Research Summary

The paper investigates the free‑boundary minimality of axial hyperplanes inside high‑dimensional circular cones. Let Ω_λ⊂ℝ^{n+1} be the open cone defined by Ω_λ={ (x,t) : t>λ|x| } where λ>0, and let E={x₁>0}. The intersection H_λ=∂E∩Ω_λ is an (n‑1)‑dimensional surface that meets the cone’s lateral boundary S_λ=∂Ω_λ{0} orthogonally along the axis. The main result (Theorem 1.1) states that for dimensions n≥4 and for λ not exceeding the explicit threshold

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