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

2008/11/07 by Maria João Ferreira, Ferreira, Maria João, Bruno Ascenso Simões +3
Mathematics · #53C43 #58E20 #Advanced Algebra and Geometry #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.0811.1125

openalex publication_date 2008/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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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