Analysis of singularities of area-minimizing currents, Part III: branch points of planar frequency $\neq$ 2, higher order asymptotics, and the local topology

This is the third part in a series of papers developing a new framework to study the local structure of $n$-dimensional area-minimizing rectifiable currents $T$ of codimension $\geq 2$.

Parts I and II introduced an intrinsic frequency function for $T$ -- planar frequency -- and used its monotonicity properties, among other things, to establish that ${\mathcal H}^{n-2}$-a.e. branch point is a rapid-decay branch point where the planar frequency is at least $1 + \alpha$.

This paper analyses branch points of planar frequency $\neq 2$.

It establishes: (1) higher order asymptotics: at ${\mathcal H}^{n-2}$-a.e. such point, the current admits an expansion of finite order $>1$, with precise decay estimates for the remainder term; (2) branch set decomposition: the set of such branch points locally decomposes into finitely many pairwise disjoint, locally $n-2$ rectifiable sets (of locally finite measure); (3) topological control: near any branch point satisfying a specific planar-frequency criterion, the support of $T$ is homeomorphic to an $n$-dimensional disk and admits a $C^{1, \mu}$ parametrization.

(Classical complex algebraic examples show that when this frequency criterion fails, the current need not be locally homeomorphic to an $n$-disk).

The work here (as well as in parts I & II) avoids the use of center manifolds -- a technically demanding foundational component of the classical Almgren framework -- and uses instead intrinsic geometric arguments based on the monotonicity formula for planar frequency.

In part IV, a center manifold is utilised to analyse planar frequency 2 points, where the center manifold becomes necessary and geometrically canonical, satisfying additional simplifying properties.

Reduced reliance on center manifolds in our framekwork is necessitated by the structural results it establishes for $T$.