Almgren-minimality of unions of two almost orthogonal planes in ⁴

In this article, we prove that the union of two almost orthogonal planes in ⁴ is Almgren-minimal. This gives an example of a one-parameter family of minimal cones, which is a phenomenon that does not exist in ³. This work is motivated by an attempt to classify the singularities of 2-dimensional Almgren-minimal sets in ⁴. Note that the traditional methods for proving minimality (calibrations and slicing arguments) do not apply here, we are obliged to use some more complicated arguments such as a stopping time argument, harmonic extensions, Federer-Fleming projections, etc., which are rarely used to prove minimality (rather they are often used to prove regularity). The regularity results for 2-dimensional Almgren minimal sets [G. David, 'Hölder regularity of two-dimensional almost-minimal sets in ⁿ', Ann. Fac. Sci. Toulouse XVIII (2009) 65-246; G. David, 'C ¹⁺-regularity for two-dimensional almost-minimal sets in ⁿ.' J. Geom. Anal. 20 (2010) 837-954] are also needed here.