On the boundary behavior of mass-minimizing integral currents

Let Sigma be a smooth Riemannian manifold, Gamma subset of Sigma a smooth closed oriented submanifold of codimension higher than 2 and T an integral area-minimizing current in S which bounds G. We prove that the set of regular points of T at the boundary is dense in Gamma. Prior to our theorem the existence of any regular point was not known, except for some special choice of Sigma and Gamma. As a corollary of our theorem we answer to a question in Almgren's Almgren's big regularity paper from 2000 showing that, if Gamma is connected, then T has at least one point p of multiplicity 1 2, namely there is a neighborhood of the point p where T is a classical submanifold with boundary G; we generalize Almgren's connectivity theorem showing that the support of T is always connected if G is connected; we conclude a structural result on T when G consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when Sigma = Rm+1 and T is m-dimensional.