Quantitative stratification and optimal regularity for harmonic almost complex structures

TL;DR

This paper provides a simplified proof of partial regularity and extends the regularity results for energy minimizing harmonic almost complex structures by employing quantitative stratification, leading to improved regularity and rectifiability of singular sets.

Contribution

It offers a new, simpler proof of partial regularity and applies quantitative stratification to establish rectifiability and optimal regularity for harmonic almost complex structures.

Findings

Simplified proof of partial regularity

Rectifiability of singular stratum

Optimal regularity results

Abstract

In a recent interesting work [15], W.Y. He established the important partial regularity theory and the almost optimal higher regularity theory for energy minimizing harmonic almost complex structures. Based on a new observation on the structure of equations, we give an easier new proof of the partial regularity theorem, and adapting the powerful quantitative stratification method of Naber-Valtorta [22], we further prove the rectifiability of singular stratum of energy minimizing harmonic almost complex structures. Based on this, we establish an optimal regularity theory, which improves the corresponding result of He.