A generalisation of the Hopf Construction and harmonic morphisms into $\s^2$

TL;DR

This paper introduces a new family of harmonic morphisms from a 5-dimensional manifold into the 2-sphere, extending classical constructions through algebraic and equivariant methods.

Contribution

It generalizes the Hopf construction to produce harmonic morphisms into , with explicit algebraic descriptions and extension properties.

Findings

Constructed harmonic morphisms on a 5D manifold within an ellipsoidal hypersurface

Extended these morphisms continuously to a real algebraic variety

Utilized a generalized Hopf construction and equivariant theory

Abstract

In this paper we construct a new family of harmonic morphisms $\varphi:V^5\to\s^2$, where $V^5$ is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of $\c^4=\r^8$. These harmonic morphisms admit a continuous extension to the completion ${V^{\ast}}^5$, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.