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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Actions of loop groups on harmonic maps


Authors: M. J. Bergvelt and M. A. Guest
Journal: Trans. Amer. Math. Soc. 326 (1991), 861-886
MSC: Primary 58E20; Secondary 22E67
MathSciNet review: 1062870
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Abstract: We describe a general framework in which subgroups of the loop group $ \Lambda G{l_n}\mathbb{C}$ act on the space of harmonic maps from $ {S^2}$ to $ G{l_n}\mathbb{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 $ \Lambda G{l_n}\mathbb{C}$, 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".


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1062870-5
PII: S 0002-9947(1991)1062870-5
Article copyright: © Copyright 1991 American Mathematical Society