[NTNU MATH-CAG-MSRC Jointed Seminar on Geometric Analysis] 【10月25日】Brian Krummel/Analysis of singularities of area minimizing currents

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 at most $n-2$. 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 at most $n-2$ 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) in which 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 $\mathcal{S}$. We introduce a new intrinsic planar frequency function for $T$ relative to a plane, which satisfies an approximate monotonicity property and is $\leq 1$ whenever $T$ is a cone for which the planar frequency function at the origin is well-defined. The planar frequency function plays a fundamental role in showing that the singular set decomposes into the sets $\mathcal{B}$ and $\mathcal{S}$ and in the analysis of the fine structure of $\mathcal{B}$. We ultimately prove that the singular set of $T$ is countably $(n-2)$-rectifiable.