A global regularity theory for shpere-valued fractional harmonic maps

TL;DR

This paper develops a comprehensive regularity theory for sphere-valued fractional harmonic maps, providing quantitative estimates of their singular sets based on advanced stratification and differentiation techniques.

Contribution

It introduces a global regularity framework for fractional harmonic maps into spheres, extending previous partial regularity results with quantitative stratification methods.

Findings

Quantitative stratification of singular sets established

Global regularity estimates derived for fractional harmonic maps

Extension of partial regularity theory to a comprehensive framework

Abstract

In this paper we consider sphere-valued stationary/minimizing fractional harmonic mappings introduced in recent years by several authors, especially by Millot-Pegon-Schikorra \cite{Millot-Pegon-Schikorra-2021-ARMA} and Millot-Sire \cite{Millot-Sire-15}. Based on their rich partial regularity theory, we establish a quantitative stratification theory for singular sets of these mappings by making use of the quantitative differentiation approach of Cheeger-Naber \cite{Cheeger-Naber-2013-CPAM}, from which a global regularity estimates follows.