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Characterizing harmonic immersions of surfaces with indefinite metric.

Harmonic maps X:(S,h) --> N from a 2-manifold S with indefinite metric h to a semi-Riemannian manifold N are characterized, assuming that the induced metric I is nondegenerate. Except in one very special case, the characterizations involve a canonically determined holomorphic quadratic differential on a naturally chosen conformal structure. This is surprising because the Euler-Lagrange equation that X must satisfy is basically the wave equation. The Gauss map of a spacelike or timelike surface in Minkowski 3-space is shown to be harmonic if and only if mean curvature is constant. Finally, it is noted that a harmonic map X:(S,h) --> N with indefinite h and nondegenerate I normally gives rise to a sine-Gordon, a sinh-Gordon, or a cosh-Gordon equation, provided that the intrinsic curvature of I is constant.