Abstract

The use of multi-valued functions in analysing the singularities of minimal submanifolds is well-established. They were used by Almgren, for example, in estimating the size of the singular set of an area-minimizing current and more recently by Wickramasekera in work describing the branch points of stable, minimal hypersurfaces. Despite progress in these contexts, gaining precise descriptions of the singularities of minimal (i.e. `stationary’ , but not necessarily stable or area-minimizing) submanifolds is still difficult and many fundamental questions are open.

In this talk I will describe some recent results on the regularity and singularity theory of minimal two-valued Lipschitz graphs in arbitrary codimension. In codimension one, there is something like classical elliptic regularity in that a two-valued Lipschitz function whose graph is minimal must automatically be $C^{1,\alpha}$ (as a two-valued function). Naturally, in higher codimension things are more complicated and the focus is on describing the local asymptotic nature of the graph close to singular points.