Partial regularity for fractional harmonic maps into spheres

TL;DR

This paper proves that fractional harmonic maps into spheres are smooth outside a small singular set and estimates the size of this set, advancing understanding of their regularity properties.

Contribution

It establishes partial regularity results for fractional harmonic maps into spheres, including smoothness away from singularities and bounds on the singular set's Hausdorff dimension.

Findings

Fractional harmonic maps are smooth outside a small singular set.

The Hausdorff dimension of the singular set is estimated based on s and minimality.

Regularity results hold in arbitrary dimensions.

Abstract

This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.