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

Abstract

This is the first paper in a series developing a new framework for $n$-dimensional area-minimizing rectifiable currents $T$ of codim. $\ge 2$. 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 $\le 1$ on cones. These properties imply a \emph{decomposition theorem} for the singular set, which (roughly speaking) asserts the following: for any integer $q\ge 2$, the set of density $q$ singularities decomposes as ${\mathrm{sing}}_{q}T=\mathcal{S}\cup \mathcal{B}$ 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 $Z^{\prime }\in {\mathbf{B}}_{{\rho }_Z}(Z)\cap \mathrm{spt}T$ with density $\ge q$ 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 ${\mathrm{sing}}_{q}T$ 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.

84 pages; expanded intro and new Section 2. This new section provides a comparison between our approach and that of De Lellis--Minter--Skorobogatova in their contemporaneous and independent work. This parallel work establishes, by fundamentally different methods, two of the main results reached in our framework: a.e. uniqueness of tangent cones and countable rectifiability of the singular set