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 () be open and be a pseudo-Riemannian manifold. Assume that a map is and harmonic on for some , and that is also continuous across . Then is and harmonic on .
The proof uses a maximum principle, which implies uniqueness of harmonic maps that agree with on the boundary of a small ball around . Existence of a smooth harmonic map with these boundary values follows from a fixed point argument; and by uniqueness it must coincide with on the ball. An explicit (even Riemannian) example shows that the assumption of continuity of across cannot be removed from the theorem.