Dressing orbits of harmonic maps

Details Dressing orbits of harmonic maps Dressing orbits of harmonic maps

1994-10-04

arxiv.org/abs/dg-ga/9410001v1

Duke Math. J. 80 1995 353-382

Mathematics

Differential Geometry

28 pages, AmSLaTeX 1.1

Scientific paper

We study the harmonic map equations for maps of a Riemann surface into a Riemannian symmetric space of compact type from the point of view of soliton theory. There is a well-known dressing action of a loop group on the space of harmonic maps and we discuss the orbits of this action through particularly simple harmonic maps called {\em vacuum solutions}. We show that all harmonic maps of semisimple finite type (and so most harmonic $2$-tori) lie in such an orbit. Moreover, on each such orbit, we define an infinite-dimensional hierarchy of commuting flows and characterise the harmonic maps of finite type as precisely those for which the orbit under these flows is finite-dimensional.

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