All harmonic 2-spheres in the unitary group, completely explicitly

Details All harmonic 2-spheres in the unitary group, completely explicitly All harmonic 2-spheres in the unitary group, completely explicitly

2008-11-07

arxiv.org/abs/0811.1125v3

Mathematics

Differential Geometry

Some minor corrections made; one reference added; one removed

Scientific paper

We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections and avoiding the usual dbar-problems or loop group factorizations. We interpret our constructions using Segal's Grassmannian model, giving an explicit factorization of the algebraic loop group, and showing how to obtain harmonic maps into a Grassmannian.

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