#### Vol. 299, No. 1, 2019

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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
##### 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.

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##### 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
##### Authors
 Jiayu Li School of Mathematical Sciences University of Science and Technology of China Hefei China Lei Liu Max Planck Institute for Mathematics in the Sciences Leipzig Germany