The Structure of Stable Codimension One Integral Varifolds near Classical Cones of Density $Q+1/2$

Abstract: For each positive integer $Q\in\mathbb{Z}{\geq 2}$, we prove a multi-valued $C^{1,\alpha}$ regularity theorem for varifolds in the class $\mathcal{S}_Q$, i.e., stable codimension one stationary integral $n$-varifolds which have no classical singularities of vertex density $<Q$, which are sufficiently close to a stationary integral cone comprised of $2Q+1$ half-hyperplanes (counted with multiplicity) meeting along a common axis. Such a result furthers the understanding of the local structure about singularities in the (possibly branched) varifolds in $\mathcal{S}_Q$ achieved by the author and N.~Wickramasekera (\cite{minterwick}) and generalises the authors' previous work in the case $Q=2$ (\cite{minter-5-2}) to arbitrary $Q\in \mathbb{Z}{\geq 2}$. One notable difference with previous works is that our methods do not need any a priori size restriction on the (density $Q$) branch set to rule out density gaps.