Quantitative stratification and higher regularity for biharmonic maps

TL;DR

This paper establishes quantitative regularity and sharp $L^p$ bounds for biharmonic maps, demonstrating improved regularity results and uniform bounds on singularities, with extensions to special target manifolds.

Contribution

It provides the first sharp, dimension-independent $L^p$ bounds for derivatives of biharmonic maps without small energy assumptions.

Findings

Minimizing biharmonic maps are in $W^{4,p}$ for all $1  p<5/4.

Uniform bound on the number of singular points in 5D.

Extension of results to maps into special targets.

Abstract

In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in $W^{4,p}$ for all $1\le p<5/4$. Further, for minimizing biharmonic maps from $\Omega \subset \mathbb{R}^5$, we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps into special targets.