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 act on the space of harmonic maps from to . 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 , 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 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

Article copyright:
© Copyright 1991
American Mathematical Society