Abstract

In the 1960’s, Almgren developed a min-max theory for constructing weak critical points of the area functional in arbitrary closed Riemannian manifolds. The regularity theory for these weak solutions (known as stationary integral varifolds) has been a fundamental open question in geometric analysis ever since. The primary difficulty arises from the possibility of a type of degenerate singularity, known as a branch point, being present in the varifold. Allard (1972) was able to prove that the branch points form a closed nowhere dense subset; however, nothing is known regarding its size or local structure.

In this talk we will discuss recent work (joint with N. Wickramsekera) regarding what can be said about the local structure at a branch point. More precisely, we prove local structural results about branch points in a large class of stationary integral varifolds: those which are codimension one, stable, and do not contain certain so-called classical singularities. These results are directly applicable to area minimising hypersurfaces mod p, and resolve an old question from the work of B. White in this setting.