Partial regularity of harmonic maps from a Riemannian manifold into a Lorentzian manifold

Jiayu Li and Lei Liu

Vol. 299 (2019), No. 1, 33–52

DOI: 10.2140/pjm.2019.299.33

Abstract

We study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $\left(u,v\right)$ from a smooth bounded open domain $\Omega \subset {ℝ}^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $\left(m-2\right)$-dimensional Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic spheres $S^l$, $l=2,\dots ,m-1$, we show $\left(u,v\right)$ is smooth.

Keywords

Lorentzian harmonic map, stationary, partial regularity, blow-up

Mathematical Subject Classification 2010

Primary: 53C43, 58E20

Milestones

Received: 4 August 2017

Revised: 15 May 2018

Accepted: 24 June 2018

Published: 18 April 2019