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 $<Q$. For the class $\mathcal{S}_Q$ our two main results describe: (i) the nature of $V\in\mathcal{S}_Q$ when $V$ is close to a flat disk of multiplicity $Q$ (so in particular a description of $V$ locally about branch points of density $Q$); (ii) the nature of $V\in\mathcal{S}_2$ when $Q$ is close to a classical cone of density 5/2.

Our first result, taken with $Q=p/2$, is readily applicable to codimension one rectifiable area minimising currents mod $p$ for any integer $p\geq 2$, establishing local structural properties of such a current $T$ as consequences of little information, namely the (easily checked) stability of the regular part of $T$ and the fact that such a 1-dimensional singular (representative) current in $\mathbb{R}^2$ consists of $p$ rays meeting at a point. Specifically, it follows from (i) that, for even $p$, if $T$ has one tangent cone at an interior point $y$ equal to an (oriented) hyperplane $P$ of multiplicity $p/2$, then $P$ is the unique tangent cone at $y$, and $T$ near $y$ is given by the graph over $P$ of a $\frac{p}{2}$-valued function with $C^{1,\alpha}$ regularity in a certain generalised sense; this settles a basic remaining open question in the study of the local structure of codimension one area minimising currents mod $p$ near points with planar tangent cones, extending the cases $p=2$ and $p=4$ of the result (with classical $C^{1,\alpha}$ conclusions near 𝑦) which have been known since the 1970's from the De Giorgi--Allard regularity theory ([All72]) and the structure theory of White ([Whi79]) respectively. The implication to mod $p$ minimising currents of the structure theory for $\mathcal{S}_{p/2}$ is analogous to how the regularity theory for codimension one integral currents is a direct corollary of the regularity theory ([Wic14]) for $\mathcal{S}_\infty = \cap_Q\mathcal{S}_Q$ (the class of stable codimension one integral varifolds with no classical singularities).

Our second result, (ii), is the first result of its kind for non-flat cones with multiplicity $>1$ 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 $n-2$.