1996/03/13 by Paul Baird, P. Baird, J. C. Wood +3
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #dg-ga #math.DG
paper · pdf · doi:10.48550/arxiv.dg-ga/9603005
30 pages, Latex 2.09, one figure
arxiv created 1996/03/13 · openalex publication_date 1996/03/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.