Harmonic two-spheres in compact symmetric spaces, revisited

TL;DR

This paper revisits harmonic two-spheres in compact symmetric spaces, extending Uhlenbeck's uniton number bound to all compact groups and providing new methods for classification and explicit formulas.

Contribution

It introduces Morse theory-inspired methods to generalize the uniton number bound and offers new proofs and formulas for harmonic 2-spheres in symmetric spaces.

Findings

Extended uniton number bound to all compact groups

Derived Weierstrass formulas for harmonic maps

Simplified classification proofs for harmonic 2-spheres

Abstract

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.