Paul Baird and John C. Wood: Harmonic morphisms from Minkowski space and hyperbolic numbers, p.195-209

Abstract:

We show that all harmonic morphisms from $3$-dimensional Minkowski space with values in a surface have a Weierstrass representation involving the complex numbers or the hyperbolic numbers depending on the signature of the codomain. We deduce that there is a non-trivial globally defined submersive harmonic morphism from Minkowski $3$-space to a surface, in contrast to the Riemannian case. We show that a degenerate harmonic morphism on a Minkowski space is precisely a null real-valued solution to the wave equation, and we find all such.

Key Words: harmonic morphism, harmonic map, wave equation, hyperbolic number

2000 Mathematics Subject Classification: Primary: 58E20,
Secondary: 53C43.

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