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 is 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.