On the Boundary Behavior of Mass-Minimizing Integral Currents

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