The monumental work of Almgren in the early 1980s showed that the singular set of a locally area minimizing rectifiable current $T$ of dimension $n$ and codimension $\geq 2$ has Hausdorff dimension at most $n-2$.  In contrast to codimension 1 area minimizers (for which it had been established a decade earlier that the singular set has Hausdorff dimension at most $n-7$), the problem in higher codimension is substantially more complex because of the presence of branch point singularities, i.e. singular points where one tangent cone is a plane of multiplicity 2 or larger. Almgren's lengthy proof (made more accessible and technically streamlined in the much more recent work of De Lellis--Spadaro) showed first that the non-branch-point singularities form a set of Hausdorff dimension at most $n-2$ using an elementary argument based on the tangent cone type at such points, and developed a powerful array of ideas to obtain the same dimension bound for the branch set separately. In this strategy, the exceeding complexity of the argument to handle the branch set stems in large part 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). In this approach, the set of singularities (of a fixed integer density $q$) is decomposed not as branch points and non-branch-points, but as a set ${\mathcal B}$ of branch points where $T$ decays towards a (unique) plane faster than a fixed exponential rate, and the complementary set ${\mathcal S}$. The set ${\mathcal S}$ contains all (density $q$) non-branch-point singularities, but a priori it could also contain a large set of branch points. To analyse ${\mathcal S}$, the work introduces a new, intrinsic frequency function for $T$ relative to a plane, called the planar frequency function. The planar frequency function satisfies an approximate monotonicity property, and takes correct values (i.e. $\leq 1$) whenever $T$ is a cone (for which planar frequency is defined) and the base point is the vertex of the cone.  These properties of the planar frequency function together with relatively elementary parts of Almgren’s theory (Dirichlet energy minimizing multivalued functions and strong Lipschitz approximation) imply that $T$ satisfies a key approximation property along $S$: near each point of ${\mathcal S}$ and at each sufficiently small scale, $T$ is significantly closer to some non-planar cone than to any plane. This property together with a new estimate for the distance of $T$ to a union of non-intersecting planes and the blow-up methods of Simon and Wickramasekera imply that $T$ has a unique non-planar tangent cone at $\mathcal{H}^{n-2}$-a.e. point of $\mathcal{S}$ and that ${\mathcal S}$ is $(n-2)$-rectifiable with locally finite measure. Analysis of ${\mathcal B}$ using the planar frequency function and the locally uniform decay estimate along ${\mathcal B}$ recovers Almgren’s dimension bound for the singular set of $T$ in a simpler way, and (again via Simon and Wickramasekera blow-up methods) shows that ${\mathcal B}$ (and hence the entire singular set of $T$) is countably $(n-2)$-rectifiable with a unique, non-zero multi-valued harmonic blow-up at $\mathcal{H}^{n-2}$-a.e. point of ${\mathcal B}$.