Harmonic two-spheres in compact symmetric spaces, revisited

Uhlenbeck introduced an invariant, the (minimal) uniton number, of harmonic 2-spheres in a Lie group G and proved that when G=SU(n) the uniton number cannot exceed n-1. In this paper, using new methods inspired by Morse Theory, we explain this result and extend it to an arbitrary compact group G. The same methods also yield Weierstrass formulae for these harmonic maps and simple proofs of most of the known classification theorems for harmonic 2-spheres in symmetric spaces.