A key open question in geometric measure theory concerns the optimal regularity conclusion for stationary integral varifolds. The primary difficulty for this lies in understanding branch points, namely non-immersed singular points where one tangent cone is a plane with multiplicity at least 2. Both the uniqueness of such tangent cones and the optimal dimension bound are not known (the latter is known for area minimising currents, having been settled by the monumental work of Almgren).

 

In this talk, I will discuss recent work with Spencer Becker-Kahn and Neshan Wickramasekera concerning these questions, in which we show that a simple topological structural condition on the varifold in “flat density gaps” is sufficient to prove that the local structure about density 2 branch points is given by a 2-valued function (with a regularity estimate). This is a consequence of a more general epsilon-regularity theorem, akin to Allard’s regularity theorem except in a multiplicity 2 setting.