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

Abstract

In this paper, we will 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 $\Omega \subset {\mathbb{R}}^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m-2)$-dimension Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic sphere $S^l$, $l=2,...,m-1$, we will show $(u,v)$ is smooth.

References in corpus (2)

• Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space
• A global weak solution to the Lorentzian harmonic map flow