A note on the spectral deformation of harmonic maps from the two-sphere into the unitary group

TL;DR

This paper provides an explicit construction of all harmonic maps from the two-sphere into the unitary group, building on a formula involving meromorphic functions, and clarifies how these maps relate to isotropic harmonic maps through Morse theory.

Contribution

It explicitly describes the process of generating all harmonic maps from the two-sphere into the unitary group from isotropic ones using a previously established formula.

Findings

Explicit formula for harmonic maps from 2-spheres to U(n).

Method to construct all harmonic maps from isotropic ones.

Clarification of the relationship between harmonic maps and Morse theory clusters.

Abstract

In [5], together with J. C. Wood, the authors gave a completely explicit formula for all harmonic maps from $2$-spheres to the unitary group $U(n)$ in terms of freely chosen meromorphic functions on $S^2$. The simplest harmonic maps are the isotropic ones. Using Morse theory Burstall and Guest [1] showed that the harmonic maps come in clusters labeled by the isotropic ones. In this work, using the formula for harmonic maps aforementioned, we describe explicitly this procedure, showing how all harmonic maps can be built from the isotropic ones. [1] F.~E.\ Burstall and M.~A.\ Guest, \textit{Harmonic two-spheres in compact symmetric spaces, revisited}, Math. Ann. 309 (1997) 541--572. [5] M.~J. Ferreira, B.~A Sim\~oes and J.~C. Wood \emph{All harmonic $2$-spheres in the unitary group, completely explicitly}, Math. Z. {\bf 266} (2010), 953--978.