Analysis and Mathematical Physics

The presence of branch points and so-called density gaps heavily complicates the regularity theory for minimal submanifolds or, more precisely, stationary integral varifolds. Density gaps illustrate a mismatched symmetry between nearby varifolds, meaning that one cannot in general hope to infer from a tangent cone the local singularity structure of a stationary integral varifold. Branch points are a highly degenerate situation, where singularities disappear in the limiting tangent cone and are replaced with regions of higher $(i.e., >1)$ multiplicity. It is still an open question as to whether the set of branch points can have positive measure within the support of a stationary integral varifold. In many of the regularity results known for certain classes of stationary integral varifolds, branch points are either ruled out a priori, or, after significant work, are ruled out a posteriori. When density gaps may occur, usually one can prove that the singular set in a density gap region is small and use this to somehow rule out density gaps. In this talk, I will discuss recent work concerning a regularity question where both branch points can occur and there is no size assumption on the singular set within a gap region. The main result is that, for a large class of stationary integral varifolds which have stable smoothly embedded parts, near certain so-called "classical cones", a varifold is a multi-valued generalised-$C^{1,\alpha}$ perturbation of the classical cone. In particular, if such a cone arises as a tangent cone to the varifold, then the points of density equal to that of the cone form a $C^{1,\alpha} $submanifold. This work builds on earlier work by the speaker and Neshan Wickramasekera concerning the local structure of branched stable minimal hypersurfaces.