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

Here, in \cite{KrumWicb} and in \cite{KrumWicc} we study the nature of an $n$-dimensional locally area minimising rectifiable current $T$ of codimension $\ge 2$ 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 $\le 1$ on any cone ($\mathrm{\ne }P$) 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 $q\ge 2$, the set ${\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}_{q}T$ of density $q$ singularities of $T$ can be written as ${\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}_{q}T=\mathcal{S}\cup \mathcal{B}$ 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 $Z^{\mathrm{\prime }}\in {\mathbf{B}}_{{\rho }_Z}(Z)\cap \mathrm{s}\mathrm{p}\mathrm{t}T$ with density $\ge q$ and at any scale ${\rho }^{\mathrm{\prime }}<{\rho }_Z$, $T$ is significantly closer to some non-planar cone ${\mathbf{C}}_{Z^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }}}$ than to any plane, and (II) $\mathcal{B}$ is relatively closed in ${\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}_{q}T$ 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}.