Abstract
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 \({\bar{\partial}}\) -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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Ferreira, M.J., Simões, B.A. & Wood, J.C. All harmonic 2-spheres in the unitary group, completely explicitly. Math. Z. 266, 953–978 (2010). https://doi.org/10.1007/s00209-009-0607-7
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DOI: https://doi.org/10.1007/s00209-009-0607-7