On the Boundary Behavior of Mass-Minimizing Integral Currents

Abstract

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