Communications in Analysis and Geometry

Volume 27 (2019)

Number 1

Quantitative regularity for $p$-harmonic maps

Pages: 111 – 159



Aaron Naber (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)

Daniele Valtorta (Institut für Mathematik, Mathematisch-naturwissenschaftliche Fakultät, Universität Zürich, Switzerland)

Giona Veronelli (Département de Mathématiques, Université Paris 13, Villetaneuse, France)


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 $\mathcal{S}(u)=\lbrace x \: s.t.\: u \textrm{ is not continuous at } x \rbrace$, 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 $u$. In this article, we generalize the study carried out in [4] 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 [10].

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.

The first author has been supported by NSF grant DMS-1406259. The second author has been supported by SNSF projects 149539, 157452 and 200021 159403/1. The second and third authors have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Received 7 October 2014

Published 7 May 2019