New Minimal Surfaces of the Sphere $S^4$ and the Hyperbolic Space $H^4$ via Harmonic Morphisms

TL;DR

This paper introduces a novel method using harmonic morphisms to construct explicit minimal submanifolds of codimension two in spheres and hyperbolic spaces, specifically in $S^4$ and $H^4$, expanding the set of known examples.

Contribution

The paper presents the first explicit constructions of minimal submanifolds in $S^4$ and $H^4$ using complex-valued harmonic morphisms, providing new tools for geometric analysis.

Findings

Explicit examples of harmonic morphisms on $S^4$ and $H^4$

Construction of minimal submanifolds of codimension two

New method based on harmonic morphisms

Abstract

In this work we introduce a new method for the construction of minimal submanifolds of codimension two in even dimensional spheres and hyperbolic spaces. This is based on the theory of complex-valued harmonic morphisms. This gives the first explicit examples of such maps defined on the sphere $S^4$ and the hyperbolic space $H^4$.