Harmonic morphisms on heaven spaces

TL;DR

This paper demonstrates that any analytic horizontally conformal submersion from a 3D conformal manifold to a 2D conformal manifold can be locally extended to a unique harmonic morphism from the associated heaven space.

Contribution

It establishes a unique extension of horizontally conformal submersions to harmonic morphisms on heaven spaces, linking conformal geometry with harmonic analysis.

Findings

Extension of conformal submersions to harmonic morphisms is unique.

Harmonic morphisms can be constructed from conformal submersions via heaven spaces.

The result applies to both real and complex analytic cases.

Abstract

We prove that any (real or complex) analytic horizontally conformal submersion from a three-dimensional conformal manifold M to a two-dimensional conformal manifold N can be, locally, `extended' to a unique harmonic morphism from the heaven space of M to N.