Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are not all integrable

TL;DR

This paper demonstrates that not all Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are integrable, contrasting previous results for harmonic maps into complex projective planes, revealing new complexities in the deformation theory.

Contribution

It shows the existence of non-integrable Jacobi fields along harmonic maps into $S^3$ and $S^4$, challenging prior assumptions about their integrability.

Findings

Jacobi fields along certain harmonic 2-spheres are not all integrable.

Harmonic maps into $S^3$ and $S^4$ can have non-integrable Jacobi fields.

The space of harmonic maps into $S^3$ forms a smooth manifold with non-integrable deformations.

Abstract

In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sphere which have non-integrable Jacobi fields. This is particularly surprising in the case of the 3-sphere where the space of harmonic maps of any degree is a smooth manifold, each map having image in a totally geodesic 2-sphere.