In his monumental work in the early 1980s, Almgren showed that the singular set of an $n$-dimensional locally area minimizing submanifold $T$ has Hausdorff dimension $\leq n-2$.  The main difficulty is that higher codimension area minimizers can admit branch point singularities, i.e. singular points at which one tangent cone is a plane of multiplicity two or greater.  Almgren’s lengthy proof showed first that the set of non-branch-point singularities has Hausdorff dimension $\leq n-2$ using an elementary argument based on tangent cone type, and developed a powerful array of ideas to obtain the same dimension bound for the branch separately.  In this strategy, the exceeding complexity of the argument stems largely from the lack of an estimate giving decay of $T$ towards a unique tangent plane at a branch point.

We will discuss a new approach to this problem (joint work with Neshan Wickramasekera).  Our approach is inspired by De Giorgi’s work on codimension one area minimizing submanifolds, in which one first proves a regularity result for area minimizers close to a plane, thereby ruling out branch point singularities.  We use the relatively elementary parts of Almgren’s work (namely Lipschitz and harmonic approximation) to first prove uniqueness of tangent cones of an area minimizing submanifold $T$ at $\mathcal{H}^{n-2}$-singular point.  This involves using a new approximation monotonicity formula for an intrinsic frequency function which quantifies the rate at which $T$ approaches a plane.  We use this to prove locally uniform estimates for $T$ relative to non-planar cones and later to study the structure of the set of branch points at which $T$ decays “rapidly” towards a unique tangent plane.