Regularity for harmonic maps into certain Pseudo-Riemannian manifolds

Abstract

In this article, we investigate the regularity for certain elliptic systems without a $L^2$-antisymmetric structure. As applications, we prove some $\epsilon$-regularity theorems for weakly harmonic maps from the unit ball $B=B(m)\subset {\mathbb{R}}^m$ $(m\ge 2)$ into certain pseudo-Riemannian manifolds: standard stationary Lorentzian manifolds, pseudospheres ${\mathbb{S}}_{\nu }^n\subset {\mathbb{R}}_{\nu }^{n+1}$ $(1\le \nu \le n)$ and pseudohyperbolic spaces ${\mathbb{H}}_{\nu }^n\subset {\mathbb{R}}_{\nu +1}^{n+1}$ $(0\le \nu \le n-1)$. 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 ${\mathbb{S}}_1^n$ or the Anti-de-Sitter space ${\mathbb{H}}_1^n$ 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 ${\mathbb{S}}_{\nu }^n$ $(1\le \nu \le n)$ or ${\mathbb{H}}_{\nu }^n$ $(0\le \nu \le n-1)$, and obtain some $\epsilon$-regularity results for such generalized (weakly) harmonic maps.

to appear in J. Math. Pures Appl

References in corpus (2)

• Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space
• Local Lipschitz regularity for degenerate elliptic systems