Almgren minimality of products of simplex-skeleton cones
Abstract
In this paper, we prove that for every m,n≥ 2, the orthogonal product Sm× Sn⊂Rm×Rn is an Almgren minimal cone of dimension m+n−2, where Sk⊂Rk is the cone over the (k−2)-skeleton of a regular k-simplex. Lawlor and Morgan \cite{LM94} proved the minimality of each Sk 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× Y, the product of two 1-dimensional Y sets, proved in 2014 by the second author, simultaneously to all pairwise products Sm× Sn, 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.
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BibTeX
@misc{huang2026almgren,
title = {{Almgren minimality of products of simplex-skeleton cones}},
author = {Songsong Huang and Xiangyu Liang and Zaoyi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27845},
eprint = {2610.27845},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27845},
}
RIS
TY - UNPB
TI - Almgren minimality of products of simplex-skeleton cones
AU - Songsong Huang
AU - Xiangyu Liang
AU - Zaoyi Zhang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27845
UR - https://arxiv.org/abs/2610.27845
AB - In this paper, we prove that for every m,n≥ 2, the orthogonal product S_m× S_n⊂R^m×R^n is an Almgren minimal cone of dimension m+n−2, where S_k⊂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× Y, the product of two 1-dimensional Y sets, proved in 2014 by the second author, simultaneously to all pairwise products S_m× 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.
ER -