Harmonic morphisms, conformal foliations and shear-free ray congruences

TL;DR

This paper explores the deep geometric and twistorial connections between conformal foliations, harmonic morphisms, and shear-free ray congruences across Euclidean, Minkowski, and hyperbolic spaces, providing explicit methods to construct and analyze these structures.

Contribution

It establishes new geometric equivalences among conformal foliations, harmonic morphisms, and shear-free ray congruences, and offers explicit construction techniques for these structures.

Findings

Any real-analytic complex-valued harmonic morphism without critical points on Minkowski space corresponds to a shear-free ray congruence.

Boundary values of harmonic morphisms on hyperbolic space define conformal foliations in Euclidean space.

Explicit methods are provided to find and analyze such foliations.

Abstract

Equivalences between conformal foliations on Euclidean $3$-space, Hermitian structures on Euclidean $4$-space, shear-free ray congruences on Minkowski $4$-space, and holomorphic foliations on complex $4$-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued harmonic morphism without critical points defined on an open subset of Minkowski space is conformally equivalent to the direction vector field of a shear-free ray congruence, 2) the boundary values at infinity of a complex-valued harmonic morphism on hyperbolic $4$-space define a real-analytic conformal foliation by curves of an open subset of Euclidean $3$-space and all such foliations arise this way. This gives an explicit method of finding such foliations; some examples are given.