Almgren's regularity theorem

Remarks

Frederick Almgren proved the theorem in a very long work that was published in 2000, after his death. Camillo De Lellis and Emanuele Spadaro later gave a shorter proof. The proof studies blow-ups at points where the tangent cone is a plane with integer multiplicity $Q \ge 2$. Near such points the current is approximated by the graph of a $Q$-valued function that minimizes a Dirichlet energy. A frequency function controls the rate at which the current departs from the plane. The bound $m - 2$ is sharp in higher codimension. By Wirtinger's inequality, complex algebraic curves in $\C^2 = \R^4$ are area-minimizing, and the curve $\set{(z, w) : z^2 = w^3}$ has a singular point at the origin, a set of dimension $0 = 2 - 2$. In codimension $n = 1$ the singular set is much smaller: its dimension is at most $m - 7$, by the work of Federer and Simons.

Going deeper

For a student

In codimension one, regularity follows from De Giorgi's excess-decay method, because the tangent planes have multiplicity one. In higher codimension a tangent plane can carry multiplicity, so branching phenomena occur, and De Giorgi's method no longer applies. Almgren's theorem handles this with $Q$-valued functions and the frequency function, tools that have since spread to free boundary problems.