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

Abstract

For each positive integer $Q\in {\mathbb{Z}}_{\ge 2}$, 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 $Q\in {\mathbb{Z}}_{\ge 2}$. 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.

27 pages, comments welcome!

References in corpus (3)

• A Structure Theory for Stable Codimension 1 Integral Varifolds with Applications to Area Minimising Hypersurfaces mod p
• The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points
• The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than $1$

Cited by in corpus (2)

• Fine Structure of Singularities in Area-Minimizing Currents Mod$(q)$
• Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces