September 8, 2022
TITLE: Area-minimizing integral currents: singularities and structure
ABSTRACT: Almgren’s famous Big Regularity Paper proves that the interior singular set of any m-dimensional area-minimizing integral current T in any smooth Riemannian manifold M has (Hausdorff) dimension at most m-2. Except for the case m=2, when it was proved that interior singularities are isolated, little is known about the structure of the singular set. Moreover, a recent theorem by Liu proves that we cannot expect it to be a C1 (m-2)-dimensional submanifold (unless the ambient M is real-analytic) as in fact, it can be a fractal set of any Hausdorff dimension α ≤ m-2. On the other hand, it seems likely that it is an (m-2)-rectifiable set, i.e., that it can be covered by countably many C1 submanifolds.
In this talk, I will explain why the problem is challenging and how it can be broken down into easier pieces following a recent joint work with Anna Skorobogatova.