On Hölder regularity of the singular set of energy minimizing harmonic maps into closed manifolds
Michał Miśkiewicz · Calculus of Variations and Partial Differential Equations · 2020
Abstract 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 $${\mathbb S}^2$$ S2 , 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.