On the topological minimality of unions of planes of arbitrary dimension

Abstract

In this article we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in arXiv:1103.1468, where we proved the Almgren minimality (which is a weaker property than the topological minimality) of the union of two almost orthogonal 2 dimensional planes. On the one hand, the topological minimality is always proved by variations of calibration methods, but in this article, we give a continuous family topological minimal sets, hence calibrations cannot apply. The advantage of a set being topological minimal (compared to Almgren minimal) is that its product with $\R^n$ stays topological minimal. This leads also to finding minimal sets which are unions of non transversal (hence far from almost orthogonal) planes; On the other hand, regularity for higher dimensional minimal sets is much less clear than those of dimension 2, hence more efforts are needed for higher dimensional cases.