Branch points and multi-valued functions are familiar concepts from complex analysis. They also arise naturally in many geometric problems, for example appearing in compactness results within geometric analysis and gauge theory.

When working with minimal surfaces, particularly when the codimension is greater than 1, due to the absence of good estimates one is forced to work with suitably weak topologies in order to achieve a form of compactness. A key issue that then arises is that local structural information is lost in the limit: for instance, one does not know whether branch points can be locally described by the graph of a multi-valued function. Theoretically, topology could accumulate at a branch point or branch points could form a set of large measure in the limiting weak surface (which is a ‘varifold’ or ‘current’). The main problem is then to understand how one can recover some type of regularity and structural information for the limit, or even how to control the size of the singular set.

I will discuss recent progress on this problem, where we are able to show that a simple topological structural condition on the “sufficiently regular (and flat)” part of a stationary integral varifold is sufficient to guarantee that the local structure about density 2 branch points is given by a 2-valued function (with an a priori regularity estimate). This is a consequence of a more general epsilon-regularity theorem, akin to Allard’s regularity theorem (which is in the multiplicity 1, non-branched, setting). This is joint work with Spencer Becker-Kahn and Neshan Wickramasekera.