On Topological Singular Set of Maps with Finite 3-Energy into $S^3$

Abstract

We prove that the topological singular set of a map in $W^{1,3}(M,{\mathbb{S}}^3)$ is the boundary of an integer-multiplicity rectifiable current in $M$, where $M$ is a closed smooth manifold of dimension greater than 3 and ${\mathbb{S}}^3$ is the three-dimensional sphere. Also, we prove that the mass of the minimal integer-multiplicity rectifiable current taking this set as the boundary is a strongly continuous functional on $W^{1,3}(M,{\mathbb{S}}^3)$.