Federer and Fleming integral currents allows to solve the Plateau problem in arbitrary Riemannian manifolds in any dimension and co-dimension. Thanks to the monumental work of Almgren, recently revised by De Lellis and Spadaro, interior regularity is by now quite well understood. On the other hand, the current literature fails to provide (for the high co-dimension case) even a single regular point at the boundary unless we require rather restrictive assumptions on the ambient space. In this talk I will give an overview of the problem and show a first boundary regularity result for mass minimising currents in any co-dimension. In particular, I will show that the regular points are dense in the boundary. This, among other things, allows to provide a positive answer to a question of Almgren, namely that for connected boundary data, the solution is actually connected and of multiplicity one.
This is a joint work with C. De Lellis, J. Hirsch and A. Massaccesi.