On the Almgren minimality of the product of a paired calibrated set with a calibrated set of codimension 1 with singularities, and new Almgren minimal cones

TL;DR

This paper proves the Almgren minimality of products of certain calibrated sets with codimension 1 sets, leading to new singularities and cones in the classification of minimal sets.

Contribution

It establishes the Almgren minimality of products of paired calibrated sets with codimension 1 calibrated sets, introducing new singularities and cones.

Findings

Product of paired calibrated set and codimension 1 set is Almgren minimal.

New types of singularities for Almgren minimal sets are identified.

Examples include products involving Simons cone and cube skeletons.

Abstract

In this paper, we prove that the product of a paired calibrated set and a set of codimension 1 calibrated by a coflat calibration with small singularity set is Almgren minimal. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets--Plateau's problem in the setting of sets. In particular, a direct application of the above result leads to various types of new singularities for Almgren minimal sets, e.g. the product of any paired calibrated cone (such as the cone over the $d-2$ skeleton of the unit cube in $\R^d, d\ge 4$) with homogeneous area minimizing hypercones (such as the Simons cone).