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

## TL;DR

This paper establishes partial regularity results for stationary harmonic maps from Riemannian to Lorentzian manifolds, showing smoothness outside a small singular set and full regularity under certain topological conditions.

## Contribution

It proves that stationary harmonic maps into Lorentzian manifolds are smooth outside a measure-zero set and fully smooth if the target admits no harmonic spheres.

## Key findings

- Harmonic maps are smooth outside a set of Hausdorff measure zero.
- Full regularity is achieved if the target manifold lacks harmonic spheres.
- The results extend partial regularity theory to Lorentzian target manifolds.

## 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 $\Omega\subset\R^m$ 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.