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Energy minimizing harmonic maps between manifolds are known to be smoothoutside a rectifiable set of codimension $3$, called the singular set. Thepossibility that this set is not a manifold, but has arbitrarily many smallgaps in it, is not excluded in general. Here we prove that some part of thesingular set - characterized by topological and analytic properties of tangentmaps - 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 thesingular set is a manifold - this generalizes a similar result intwo-dimensional domain, due to Hardt and Lin.

On H\"older regularity of the singular set of energy minimizing harmonic maps into closed manifolds