A key problem in geometric measure theory is to understand singularities in minimal surfaces when multiplicity occurs in the tangent cone. For questions concerning partial regularity, the primary situation to understand is that of a branch point, namely a (non-immersed) singular point where one tangent cone is a plane of multiplicity strictly greater than 1. The size of the set of such singularities, uniqueness of the tangent cone, and the local topology of the surface are all still unknown — the latter two questions are (in general) open even for area minimisers.
I will discuss joint work with Spencer Becker-Kahn and Neshan Wickramasekera which aims to improve our understanding of the nature of stationary integral varifolds near planes of multiplicity 2. In particular, we are able to prove an epsilon-regularity theorem (akin to Allard’s regularity theorem for multiplicity 1 planes, but now as a 2-valued graph), in any dimension and codimension, provided the varifold is stationary and satisfies a suitable (non-variational) structural hypothesis. In general, our theorem provides a ‘gap-or-decay’ type result for arbitrary stationary integral varifolds. The former result allows us to understand branch points of stationary 2-valued Lipschitz graphs, for instance proving uniqueness of the tangent plane at any such point and classifying nearby singularities.