## Tohoku Mathematical Journal 2006 June SECOND SERIES VOL. 58, NO. 2

 Tohoku Math. J. 58 (2006), 231-236

Title THE SPACE OF HARMONIC TWO-SPHERES IN THE UNIT FOUR-SPHERE

Author John Bolton and Lyndon M. Woodward

(Received May 26, 2004, revised April 13, 2005)
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.

2000 Mathematics Subject Classification. Primary 58D10; Secondary 53C43.

Key words and phrases. Harmonic maps, 2-sphere, twistor fibration.