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

We study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u, v)$ from a smooth bounded open domain $Ω \subset ℝ^{m}$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m - 2)$ -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 $(u, v)$ 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

Vol. 299, No. 1, 2019

Jiayu Li and Lei Liu

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors