Constant frequency and the higher regularity of branch sets

TL;DR

This paper proves that for certain two-valued energy-minimizing or harmonic functions, the singular set is a real analytic submanifold when the Almgren frequency is of the form 1/2 plus an integer, and that the frequency cannot be an integer.

Contribution

It establishes that the Almgren frequency at singular points is always of the form 1/2 plus an integer and proves the real analyticity of the singular set in this case.

Findings

Frequency at singular points is always 1/2 + k for some integer k.

Singular set is a C^{1,τ} submanifold under certain conditions.

Singular set is real analytic when frequency equals 1/2 + k.

Abstract

We consider a two-valued function $u$ that is either Dirichlet energy minimizing, $C^{1,\mu}$ harmonic, or in $C^{1,\mu}$ with an area-stationary graph such that Almgren's frequency (restricted to the singular set) is continuous at a singular point $Y_0$. As a corollary of recent work of Wickramasekera and the author, if the frequency of $u$ at $Y_0$ equals $1/2+k$ for some integer $k \geq 0$, then the singular set of $u$ is a $C^{1,\tau}$ submanifold and we have estimates on the asymptotic behavior of $u$ at singular points. Using a nontrivial modification of the argument of Wickramasekera and author, we show that the frequency of $u$ at $Y_0$ cannot equal an integer and therefore must equal $1/2+k$ for some integer $k \geq 0$. We then use the asymptotic behavior of $u$ and partial Legendre-type transformations based on those of Kinderlehrer, Nirenberg, and Spruck to show that the singular set in this case is in fact real analytic.