At the end of his monograph Almgren addresses the question of regularity of solutions at the boundary. Full regularity was proved by Allard in his PhD thesis when the ambient manifold is Euclidean space and the boundary surface lies in the boundary of a uniformly convex open set. The general case in codimension 1 was then settled by Hardt and Simon in the early eighties. But in codimension higher than 1 and in general ambient manifolds the current state of the art does not even guarantee the existence of a single boundary regular point. This prevents the understanding of seemingly innocent questions like the following: does the connectedness of the boundary imply the connectedness of the minimizer?
In a joint work with Guido de Philippis, Jonas Hirsch and Annalisa Massaccesi we give a first general boundary regularity theory which allows us to answer positively to the question above.