On Hölder regularity of the singular set of energy minimizing harmonic maps into closed manifolds

Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension 3, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set — characterized by topological and analytic properties of tangent maps — is a topological manifold. In the special case of maps into the sphere S^2, we conclude that the whole top-dimensional part of the singular set is a manifold — this generalizes a similar result in four-dimensional domains, due to Hardt and Lin.