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Camillo De Lellis: Lectures

September 8, 2023
TITLE: Area-minimizing integral currents: singularities and structure

ABSTRACT: Let T be an area-minimizing integral current of dimension m in a smooth closed Riemannian manifold of dimension m + n. It is known since the work of De Giorgi, Fleming, Almgren, Simons, and Federer in the sixties and seventies that, when n = 1, the (interior) singular set of T has dimension at most m − 7. In higher codimension Almgren’s big regularity paper proved in 1980 that the singular set has dimension at most m − 2, laying the grounds for a theory which has been simplified and extended in the last 15 years. Both theorems are optimal, but at the qualitative level there is a quite important mismatch between the singular sets of the known examples and a general closed set of the same dimension. In a celebrated work in the nineties Simon proved, for n = 1, that the singular set is m − 7- rectifiable and that the tangent cone is unique Hm−7-a.e.. The counterpart of Simon’s theorem in higher codimension has been reached very recently by Paul Minter, Anna Skorobogatova and myself and, independently, by Krummel and Wickramasekera. Even though it would be natural to expect much stronger structural results, our theorem is indeed close to optimal, as a recent result of Liu shows that the singular set can in fact be a fractal of any Hausdorff dimension α ≤ m − 2.