On the Structure and Regularity of Stable Branched Minimal Hypersurfaces

Abstract

We study the structure of large classes of stable codimension one stationary integral varifolds $V$ near higher multiplicity hyperplanes and classical cones, i.e. cones supported on half-hyperplanes which have a common boundary. A key motivation for our work is to understand the local structure of stationary integral varifolds about their branch points. In addition to the hypotheses above, the only additional assumption we make is that the varifolds do not admit classical singularities of certain densities. More precisely, for any given $Q\in {3/2,2,5/2,3,\dotsc}$ we consider the class $\mathcal{S}_Q$ of codimension one stationary integral varifolds which have stable regular part and admit no classical singularities of density $1$ when branch points are present in the nearby varifold, and in particular completes the analysis of the singular set of $V\in\mathcal{S}_2$ in the region where the density is $<3$, up to a set of dimension at most $n-2$.