Codimension-three bound for singular sets of stationary harmonic maps

Determine whether the singular set of a general stationary harmonic map has Hausdorff codimension at least three.

Background

The paper explains that, because of the loss of compactness in sequences of stationary harmonic maps and the resulting defect measures, the currently established regularity theory only guarantees that the singular set has zero codimension-two Hausdorff measure. The authors state that it remains unresolved whether the stronger codimension-three bound holds for general stationary harmonic maps. This question is distinct from the paper’s main result, which proves sharper bounds for stable stationary harmonic maps into round spheres.