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

This is the first paper in a series developing a new framework for $n$-dimensional area-minimizing rectifiable currents $T$ of codim. $\geq 2$. Our approach relies on an intrinsic frequency function for $T$, the \emph{planar frequency}, introduced in the present paper. We establish that planar frequency satisfies an approximate monotonicity property, and takes values $\leq 1$ on cones. These properties imply a \emph{decomposition theorem} for the singular set, which (roughly speaking) asserts the following: for any integer $q \geq 2$, the set of density $q$ singularities decomposes 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 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 to a unique tangent plane at a rate $o(\rho^{1 + \alpha})$ as the scale $\rho \to 0$, where $\alpha$ is a locally uniform constant. This is central to the more refined analysis in the subsequent papers. The program establishes: (i) uniqueness of tangent cones at ${\mathcal H}^{n-2}$ a.e. point; (ii) singular set decomposition into fintely many disjoint, locally compact, locally $(n-2)$-rectifiable sets (of locally finite measure); (iii) $T$ admits an asymptotic expansion of finite order $> 1$ with remainder estimates at ${\mathcal H}^{n-2}$-a.e. branch point; and (iv) near any branch point satisfying a specific frequency criterion, $T$ is homeomorphic to an $n$-dimensional disk and admits a $C^{1, \mu}$ parameterization.