P. Baird et Jc. Wood, HARMONIC MORPHISMS, CONFORMAL FOLIATIONS AND SHEAR-FREE RAY CONGRUENCES, Bulletin of the Belgian Mathematical Society Simon Stevin, 5(4), 1998, pp. 549-564
A shear-free ray congruence is a foliation by null lines (light rays)
of an open subset of Minkowski space satisfying a certain conformality
condition. We show that (i) any real-analytic complex-valued harmonic
morphism without critical points defined on an open subset of Minkows
ki space is conformally equivalent to the direction vector held of a s
hear-free ray congruence, (ii) any (real-analytic) complex-valued hori
zontally conformal submersion on an open subset of R-3 is locally the
boundary values at infinity of a harmonic morphism on an open subset o
f hyperbolic space. This provides a construction of families of minima
l surfaces in hyperbolic 4-space with given boundaries at infinity.