Quantitative regularity for p-harmonic maps

TL;DR

This paper establishes quantitative regularity results for p-harmonic maps between Riemannian manifolds, providing Minkowski estimates on the singular set and analyzing both minimizing and stationary cases with improved bounds on singularities.

Contribution

It generalizes previous regularity results from 2-harmonic to p-harmonic maps and introduces quantitative stratification techniques for analyzing singularities.

Findings

Minkowski volume estimates on singular sets for p-harmonic maps.

Extension of regularity results from 2-harmonic to general p-harmonic maps.

Refined bounds on the number of isolated singularities.

Abstract

In this article, we study the regularity of minimizing and stationary $p$-harmonic maps between Riemannian manifolds. The aim is obtaining Minkowski-type volume estimates on the singular set $S(f)=\{x \ \ s.t. \ \ f \text{ is not continuous at } x\}$, as opposed to the weaker and non quantitative Hausdorff dimension bounds currently available in literature for generic $p$. The main technique used in this paper is the quantitative stratification, which is based on the study of the approximate symmetries of the tangent maps of $f$. In this article, we generalize the study carried out in \cite{ChNa2} for minimizing $2$-harmonic maps to generic $p\in (1,\infty)$. Moreover, we analyze also the stationary case where the lack of compactness makes the study more complicated. In order to understand the degeneracy intrinsic in the behaviour of stationary maps, we study the defect measure naturally associated to a sequence of such maps and generalize the results obtained in \cite{lin_stat}. By using refined covering arguments, we also improve the estimates in the case of isolated singularities and obtain a definite bound on the number of singular points. This result seems to be new even for minimizing $2$-harmonic maps.