Camillo De Lellis (IAS Princeton)

Title: “The oriented Plateau problem and a question of Almgren”

Abstract: The theory of integral currents allows to solve the Plateau problem in smooth Riemannian manifolds in any dimension and codimension. The interior regularity theory is well developed both in codimension 1 and in higher codimension, while the boundary regularity is fully resolved only in codimension 1. Indeed, in codimension higher than 1 the current literature fails to provide even a single regular point at the boundary unless we require rather restrictive assumptions on the ambient space and the boundary datum. In this talk I will give an overview of the problem and show a first boundary regularity result for mass minimizing currents in any co-dimension, any ambient manifold and at any regular boundary. This, among other things, provides a positive answer to a question of Almgren and paves the way to further developments in the area.