### Abstract

We describe a general framework in which subgroups of the loop group AGl_{n}C act on the space of harmonic maps from S^{2} to Gl_{n}C. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of AGl_{n}nC, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action (“completion”) gives rise to her fundamental process of “adding a uniton”. It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from S^{2} to complex projective space, we describe a modification of this completion procedure which does indeed reproduce “adding a uniton”.

Original language | English |
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Pages (from-to) | 1-26 |

Number of pages | 26 |

Journal | Transactions of the American Mathematical Society |

Volume | 326 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1991 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Transactions of the American Mathematical Society*,

*326*(2), 1-26. https://doi.org/10.1090/S0002-9947-1991-1062870-5