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Removability of singularities of harmonic maps into pseudo-Riemannian manifolds. (English) Zbl 1061.58014
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. 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 $\Omega$. 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.
##### MSC:
 58E20 Harmonic maps between infinite-dimensional spaces 53C50 Lorentz manifolds, manifolds with indefinite metrics
##### Keywords:
harmonic maps; removable singularity; maximum principle
Full Text:
##### References:
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