Optimal higher regularity for biharmonic maps via quantitative stratification

TL;DR

This paper improves the regularity results for biharmonic maps by applying quantitative stratification, achieving optimal regularity for minimizing biharmonic maps, building on previous work by Breiner, Lamm, Cheeger, Naber, and Valtorta.

Contribution

It refines existing regularity results for biharmonic maps using advanced stratification techniques, establishing optimal regularity for minimizing cases.

Findings

Achieved optimal regularity for minimizing biharmonic maps.

Enhanced understanding of regularity via quantitative stratification.

Built upon and improved previous regularity results.

Abstract

This little note is devoted to refining the almost optimal regularity results of Breiner and Lamm \cite{Breiner-Lamm-2015} on minimizing and stationary biharmonic maps via the powerful quantitative stratification method introduced by Cheeger and Naber \cite{Cheeger-Naber-2013} and further developed by Naber and Valtorta \cite{Naber-V-2017,Naber-V-2018} for harmonic maps. In particular, we obtain an optimal regularity results for minimizing biharmonic maps.