Harmonic morphisms and bicomplex manifolds

TL;DR

This paper introduces a unified framework for harmonic morphisms from 3D Euclidean or pseudo-Euclidean spaces to 2D surfaces using bicomplex functions, connecting complex analysis with geometric structures.

Contribution

It develops a novel approach using bicomplex variables to unify various harmonic morphism constructions and explores their geometric and conformal compactifications.

Findings

Bicomplex-holomorphic functions characterize harmonic morphisms in the studied settings.

Real slices recover classical compactifications of harmonic morphisms.

Complex-Riemannian manifolds can be interpreted as bicomplex manifolds for conformal compactifications.

Abstract

We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of complex-harmonic morphism between complex-Riemannian manifolds and showing how these are given by bicomplex-holomorphic functions when the codomain is one-bicomplex dimensional. By taking real slices, we recover well-known compactifications for the three possible real cases. On the way, we discuss some interesting conformal compactifications of complex-Riemannian manifolds by interpreting them as bicomplex manifolds.