Harmonic Maps from $S^3$ into $S^2$ with low Morse Index

TL;DR

This paper classifies low Morse index harmonic maps from the 3-sphere to the 2-sphere, showing they are essentially compositions involving the Hopf fibration and holomorphic maps.

Contribution

It proves that all harmonic maps from $S^3$ to $S^2$ with Morse index ≤ 4 are harmonic morphisms composed of isometries, the Hopf fibration, and holomorphic maps.

Findings

Harmonic maps with Morse index ≤ 4 are harmonic morphisms.

Such maps are compositions of isometries, Hopf fibration, and holomorphic maps.

The classification provides a complete description of low Morse index harmonic maps.

Abstract

We prove that any smooth harmonic map from $S^3$ into $S^2$ of Morse index less or equal than $4$ has to be an harmonic morphism, that is the successive composition of an isometry of $S^3$, the Hopf fibration and an holomorphic map from ${\mathbb C}P^1$ into itself.