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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.