On H\"older 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 ${\mathbb S}^2$, we conclude that the whole top-dimensional part of the singular set is a manifold - this generalizes a similar result in two-dimensional domain, due to Hardt and Lin.