On the Topological Minimality of Unions of Planes of Arbitrary Dimension

In this article, we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in [13], where we proved the Almgren minimality (which is a weaker property than the topological minimality) of the union of two almost orthogonal two-dimensional planes. On the one hand, the topological minimality is almost 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 with Almgren minimal) is that its product with Rⁿ stays topological minimal. This leads also to finding minimal sets which are unions of non transversal (hence far from almost orthogonal) planes. Notice that the regularity for higher dimensional minimal sets is much less clear than those of dimension 2, hence more efforts are needed for the present higher dimensional case.