Actions of loop groups on harmonic maps
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- by M. J. Bergvelt and M. A. Guest PDF
- Trans. Amer. Math. Soc. 326 (1991), 861-886 Request permission
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".References
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Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 326 (1991), 861-886
- MSC: Primary 58E20; Secondary 22E67
- DOI: https://doi.org/10.1090/S0002-9947-1991-1062870-5
- MathSciNet review: 1062870