Some configuration results for area-minimizing cones

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.