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

TL;DR

This paper provides an explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n), including the 2-sphere, using meromorphic functions and avoiding complex loop group factorizations.

Contribution

It introduces a fully explicit construction method for harmonic maps into U(n) and Grassmannians, bypassing traditional dbar-problems and loop group techniques.

Findings

Explicit formulas for harmonic maps from Riemann surfaces to U(n).

Construction of harmonic maps into Grassmannians using the new method.

Simplification of the process by avoiding loop group factorizations.

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.