Fine properties of branch point singularities: stationary two-valued graphs and stable minimal hypersurfaces near points of density $< 3$

TL;DR

This paper analyzes the asymptotic behavior of two-valued stationary graphs near branch points, revealing their structure, rectifiability, and regularity, especially for points with density less than 3, advancing understanding of minimal hypersurface singularities.

Contribution

It establishes the asymptotic decay to harmonic tangent functions at branch points and proves rectifiability and analyticity of the branch locus for stable minimal hypersurfaces with density less than 3.

Findings

Branch locus is countably $(n-2)$-rectifiable.

Near certain points, branch locus is an embedded real analytic submanifold.

Asymptotic behavior characterized by homogeneous cylindrical harmonic functions.

Abstract

We study (higher order) asymptotic behaviour near branch points of stationary $n$-dimensional two-valued $C^{1, \mu}$ graphs in an open subset of ${\mathbb R}^{n+m}$. Specifically, if $M$ is the graph of a two-valued $C^{1, \mu}$ function $u$ on an open subset $\Omega \subset {\mathbb R}^{n}$ taking values in the space of un-ordered pairs of points in ${\mathbb R}^{m}$, and if the integral varifold $V = (M, \theta),$ where the multiplicity function $\theta \, : \, M \rightarrow \{1, 2\}$ is such that $\theta =2$ on the set where the two values of $u$ agree and $\theta =1$ otherwise, is stationary in $\Omega \times {\mathbb R}^{m}$ with respect to the mass functional, we show that at ${\mathcal H}^{n-2}$-a.e.\ point $Z$ along its branch locus $u$ decays asymptotically, modulo its single valued average, to a unique non-zero two-valued cylindrical harmonic tangent function $\varphi^{(Z)}$ which is homogeneous of some degree $\geq 3/2$. As a corollary, we obtain that the branch locus of $u$ is countably $(n-2)$-rectifiable, and near points $Z$ where the degree of homogeneity of $\varphi^{(Z)}$ is equal to $3/2$, the branch locus is an embedded real analytic submanifold of dimension $n-2$. These results, combined with the recent works \cite{M} and \cite{MW}, imply a stratification theorem for the (relatively open) set of density $< 3$ points of a stationary codimension 1 integral $n$-varifold with stable regular part and no triple junction singularities.