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Regularity for harmonic maps into certain Pseudo-Riemannian manifolds

In this article, we investigate the regularity for certain elliptic systems without a $L^2$-antisymmetric structure. As applications, we prove some $ε$-regularity theorems for weakly harmonic maps from the unit ball $B= B(m) \subset \mathbb{R}^m $ $(m\geq2)$ into certain pseudo-Riemannian manifolds: standard stationary Lorentzian manifolds, pseudospheres $\mathbb{S}^n_ν\subset \mathbb{R}^{n+1}_ν$ $(1\leqν\leq n)$ and pseudohyperbolic spaces $\mathbb{H}^n_ν\subset \mathbb{R}^{n+1}_{ν+1}$ $(0\leqν\leq n-1)$. Consequently, such maps are shown to be Hölder continuous (and as smooth as the regularity of the targets permits) in dimension $m=2$. In particular, we prove that any weakly harmonic map from a disc into the De-Sitter space $\mathbb{S}^n_1$ or the Anti-de-Sitter space $\mathbb{H}^n_1$ is smooth. Also, we give an alternative proof of the Hölder continuity of any weakly harmonic map from a disc into the Hyperbolic space $\mathbb{H}^n$ without using the fact that the target is nonpositively curved. Moreover, we extend the notion of generalized (weakly) harmonic maps from a disc into the standard sphere $\mathbb{S}^n$ to the case that the target is $\mathbb{S}^n_ν$ $(1\leqν\leq n)$ or $\mathbb{H}^n_ν$ $(0\leqν\leq n-1)$, and obtain some $ε$-regularity results for such generalized (weakly) harmonic maps.