Almgren minimality of products of simplex-skeleton cones

Source abstract

In this paper, we prove that for every $m,n\geq 2$, the orthogonal product $S_m\times S_n\subset\R^m\times\R^n$ is an Almgren minimal cone of dimension $m+n-2$, where $S_k\subset\R^k$ is the cone over the $(k-2)$-skeleton of a regular $k$-simplex. Lawlor and Morgan \cite{LM94} proved the minimality of each $S_k$ by paired calibrations, while Guy David \cite{DJT, GD13} later raised the general question whether products of Almgren minimal sets remain minimal. Our theorem gives a affirmative answer to this product question within the entire Lawlor-Morgan regular-simplex family. It extends the isolated example \(\Y\times \Y\), the product of two 1-dimensional $\Y$ sets, proved in 2014 by the second author, simultaneously to all pairwise products $S_m\times S_n$, and therefore produces a large new family of possible codimension-two singularities for Plateau's problem. The key idea of the proof is to find a new way to combine homology with paired calibrations in higher codimension. Instead of using only the currents associated with the facets, we consider all their nonzero $\FF$ combinations. The resulting complete family has enough algebraic symmetry to control uniformly the overlaps created by noninjective deformations.