Actions of loop groups on harmonic maps

M. J. Bergvel, Martin Guest

Research output: Contribution to journalArticle

25 Citations (Scopus)

Abstract

We describe a general framework in which subgroups of the loop group AGlnC act on the space of harmonic maps from S2 to GlnC. 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 AGlnnC, 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 S2 to complex projective space, we describe a modification of this completion procedure which does indeed reproduce “adding a uniton”.

Original languageEnglish
Pages (from-to)1-26
Number of pages26
JournalTransactions of the American Mathematical Society
Volume326
Issue number2
DOIs
Publication statusPublished - 1991
Externally publishedYes

Fingerprint

Loop Groups
Harmonic Maps
Completion
Riemann-Hilbert Problem
Complex Projective Space
Simplification
Circle
Union
Limiting
Subgroup
Decomposition
Decompose
Energy

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Actions of loop groups on harmonic maps. / Bergvel, M. J.; Guest, Martin.

In: Transactions of the American Mathematical Society, Vol. 326, No. 2, 1991, p. 1-26.

Research output: Contribution to journalArticle

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