Transverse singularities of minimal two-valued graphs in arbitrary codimension

Abstract

We prove some epsilon regularity results for $n$-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one $n$-dimensional planes, and in a neighbourhood of any point at which at which at least one tangent cone is equal to a union of four distinct multiplicity one $n$-dimensional half-planes that meet along an $(n-1)$-dimensional axis. The key ingredient is a new Excess Improvement Lemma obtained via a blow-up method (inspired by the work of L. Simon on the singularities of ‘multiplicity one’ classes of minimal submanifolds) and which can be iterated unconditionally. We also show that any tangent cone to an $n$-dimensional minimal two-valued Lipschitz graph that is translation invariant along an $(n-1)$ or $(n-2)$-dimensional subspace is indeed a cone of one of the two aforementioned forms, which yields a global decomposition result for the singular set.