Quantitative stratification and global regularity for 1/2-harmonic mappings

TL;DR

This paper advances the understanding of 1/2-harmonic maps into manifolds by developing a quantitative stratification approach, extending regularity theory, and proving rectifiability of singular sets using harmonic extension methods.

Contribution

It adapts defect measure theory and stratification techniques to 1/2-harmonic maps, providing new regularity and singular set structure results.

Findings

Quantitative stratification of singular points.

Sharp growth estimates for tubular neighborhoods.

Rectifiability of singular strata.

Abstract

In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of {the celebrated harmonic extension method by Caffarelli-Silverstre}. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum.