Jacobi fields along harmonic 2-spheres in ${\bf C}P^2$ are integrable

Abstract

We show 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). This provides one of the few known answers to this problem of integrability, which was raised in different contexts of geometry and analysis. It implies that the Jacobi fields form the tangent bundle to each component of the manifold of harmonic maps from $S^2$ to ${\bf C}P^2$ thus giving the nullity of any such harmonic map; it also has bearing on the behaviour of weakly harmonic $E$-minimizing maps from a 3-manifold to ${\bf C}P^2$ near a singularity and the structure of the singular set of such maps from any manifold to ${\bf C}P^2$.