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

TL;DR

This paper proves that certain parts of the singular set of energy minimizing harmonic maps into manifolds are topological manifolds, with full regularity results for maps into spheres, extending previous two-dimensional results.

Contribution

It establishes that parts of the singular set characterized by tangent maps are topological manifolds, generalizing known results to higher dimensions and specific target manifolds.

Findings

Part of the singular set is a topological manifold.

The entire top-dimensional singular set for maps into spheres is a manifold.

Extends regularity results from 2D to higher dimensions.

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 ${\mathbb S}^2$, 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.