On the Structure and Regularity of Stable Branched Minimal Hypersurfaces
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Abstract
We study the structure of large classes of stable codimension one stationary integral varifolds
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 $n-2$.