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Zbl 1061.58014
Hélein, Frédéric
Removability of singularities of harmonic maps into pseudo-Riemannian manifolds. (English)
[J] Ann. Fac. Sci. Toulouse, VI. Sér., Math. 13, No. 1, 45-71 (2004). ISSN 0240-2963

A removable singularity theorem is proved for harmonic maps from a Euclidean domain to a pseudo-Riemannian manifold. A map between such spaces is called harmonic if it is a critical point of the energy functional associated with the indefinite metric of the target. Equivalently, it is harmonic if it satisfies the harmonic map equation with the Riemannian metric tensor replaced by the pseudo-Riemannian one. \par Let $\Omega\subseteq R^n$ ($n\ge2$) be open and $(N,h)$ be a pseudo-Riemannian manifold. Assume that a map $u:\Omega\to N$ is $C^2$ and harmonic on $\Omega\setminus\{a\}$ for some $a\in\Omega$, and that $u$ is also continuous across $a$. Then $u$ is $C^2$ and harmonic on $\Omega$. \par The proof uses a maximum principle, which implies uniqueness of harmonic maps that agree with $u$ on the boundary of a small ball around $a$. Existence of a smooth harmonic map with these boundary values follows from a fixed point argument; and by uniqueness it must coincide with $u$ on the ball. An explicit (even Riemannian) example shows that the assumption of continuity of $u$ across $a$ cannot be removed from the theorem.
[Andreas Gastel (Düsseldorf)]

MSC 2000:: *58E20 Harmonic maps between infinite-dimensional spaces
53C50 Lorentz manifolds, manifolds with indefinite metrics

Keywords: harmonic maps; removable singularity; maximum principle

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