Analysis of singularities of area minimizing currents: planar frequency, branch points of rapid decay, and weak locally uniform approximation

Abstract

Here, in \cite{KrumWicb} and in \cite{KrumWicc} we study the nature of an $n$-dimensional locally area minimising rectifiable current $T$ of codimension $\geq 2$ near its typical (i.e.\ ${\mathcal H}^{n-2}$ a.e.) singular points. Our approach relies on an intrinsic frequency function for $T$, which we call the \emph{planar frequency function}, which is defined geometrically relative to a given $n$-dimensional plane $P$ and a given base point in the support of $T$. In the present article we establish that the planar frequency function satisfies an approximate monotonicity property, and takes values $\leq 1$ on any cone ($\neq P$) meeting $P^{\perp}$ only at the origin, whenever the base point is the vertex of the cone. Using these properties we obtain a \emph{decomposition theorem} for the singular set of $T$, which (roughly speaking) asserts the following: for any integer $q \geq 2$, the set ${\rm sing}_{q} \, T$ of density $q$ singularities of $T$ can be written as ${\rm sing}_{q} \, T = {\mathcal S} \cup {\mathcal B}$ for disjoint sets ${\mathcal S}$ and ${\mathcal B}$, where: (I) each point $Z \in {\mathcal S}$ has a neighbourhood ${\mathbf B}_{\rho_{Z}}(Z)$ such that about any point $Z^{\prime} \in {\mathbf B}_{\rho_{Z}}(Z) \cap {\rm spt} \, T$ with density $\geq q$ and at any scale $\rho^{\prime} < \rho_{Z}$, $T$ is significantly closer to some non-planar cone ${\mathbf C}_{Z^{\prime}, \rho^{\prime}}$ than to any plane, and (II) ${\mathcal B}$ is relatively closed in ${\rm sing}_{q} \, T$ and $T$ satisfies a locally uniform estimate along ${\mathcal B}$ implying decay of $T$ to a unique tangent plane faster than a fixed exponential rate in the scale. This result is central to the more refined analysis of $T$ we perform in \cite{KrumWicb} and \cite{KrumWicc}.