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

TL;DR

This paper establishes a regularity theorem for stable codimension one integral varifolds near specific classical cones, extending previous results to all integers Q ≥ 2 without size restrictions on the branch set.

Contribution

It generalizes earlier work by proving a multi-valued C^{1,α} regularity result for varifolds close to cones with 2Q+1 half-hyperplanes, for all Q ≥ 2.

Findings

Proves regularity near classical cones with 2Q+1 half-hyperplanes

Extends previous results from Q=2 to all Q ≥ 2

Does not require size restrictions on the branch set

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.