The space of harmonic two-spheres in the unit four-sphere

Abstract

A harmonic map of the Riemann sphere into the unit 4-dimensional sphere has area $4\pi\! d$ for some positive integer $d$, and it is well-known that the space of such maps may be given the structure of a complex algebraic variety of dimension $2d+4$. When $d$ less than or equal to 2, the subspace consisting of those maps which are linearly full is empty. We use the twistor fibration from complex projective 3-space to the 4-sphere to show that, if $d$ is equal to 3, 4 or 5, this subspace is a complex manifold.