Some configuration results for area-minimizing cones

TL;DR

This paper establishes broad configuration results for constructing area-minimizing cones, showing that cones over certain minimal submanifolds and their products are area-minimizing without extra geometric assumptions.

Contribution

It proves that cones over minimal products of submanifolds and spheres are area-minimizing, expanding understanding of minimal cone configurations without additional geometric constraints.

Findings

Cones over minimal products of submanifolds are area-minimizing.

Cones over products of submanifolds and large-dimensional spheres are area-minimizing.

The categories of regular area-minimizing cones and minimal submanifolds have the same cardinality.

Abstract

We discover some very general configuration results for constructing area-minimizing cones. In particular, given any closed minimal submanifold in some Euclidean sphere, every cone over the minimal product of sufficiently many copies of the submanifold turns out to be area-minimizing; meanwhile every cone over the minimal product of the submanifold and a round sphere of sufficiently large dimension is also area-minimizing. Here no additional geometric assumption (e.g. on isometry group or second fundamental form) is required. Moreover, we establish that the category of regular area-minimizing cones in Euclidean spaces and that of closed minimal submanifolds in Euclidean spheres share the same cardinality.