Loop group actions on harmonic maps and their applications

Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems

“…The first extensive use of these ideas is due to K. Uhlenbeck in 1989 [82] and N. Hitchin in 1990 [54] and was followed by many other works [11], [12], [21], [22], [30], [41], [47], [49], [51]. The idea of loop groups, originated by works by M. Sato [74], was developped by G. Segal In all cases we have furthermore that the mapping is a diffeomorphism, a property that we shall summarize by the relation j\1C = j\.s:Bj\.…”

Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass-type representations

“…It is clear that ζ ζ l . Now, according to (11) and (12), we can take ∆ ′ ̺ = {L i − L n , L n − L i }. Hence, for i > 0,…”

“…Theorem 10. [8,11] Given ξ ∈ I(G) ∩ k σ , any harmonic map ϕ : S 2 → P σ ξ ⊂ G admits an T σ -invariant extended solution Φ : S 2 → Ω σ G. Conversely, given an T σ -invariant extended solution Φ, the smooth map ϕ = Φ −1 from S 2 is harmonic and takes values in some connected component of P σ .…”

“…The theory of Burstall and Guest [1] on harmonic two-spheres in compact inner symmetric G-spaces was extended by Eschenburg, Mare and Quast [8] to outer symmetric spaces as follows: each harmonic map from a two-sphere into an outer symmetric space G/G σ , with outer involution σ, corresponds to an extended solution Φ which is invariant under a certain involution T σ induced by σ on ΩG (see also [11]); Φ takes values in some unstable manifold U ξ , off some discrete set; under the gradient flow of the energy any such invariant extended solution is deformed to an extended solution with values in some conjugacy class of a Lie group homomorphism γ ξ : S 1 → G σ ; applying the normalization procedure of extended solutions introduced by Burstall and Guest for Lie groups, ξ can be chosen among the canonical elements of I(G σ ) I(G); if G has trivial centre, there are precisely 2 k canonical homorphisms, where k = rank(G σ ) < rank(G); hence there are at most 2 k classes of harmonic two-spheres in G/G σ if G has trivial centre.…”

Group actions and deformations for harmonic maps into symmetric spaces

“…The embedding t : G/C P = G C /G P -> Ω'G = L'G C /L+>'G C is holomorphic and G c -equivariant with respect to the injective homomorphism G c -» L'G C of constant loops. Then the action of G on G/Cp preserves the holomorphicity condition and the super-horizontality condition, and the action of K c on G/Cp preserves the holomorphicity condition and the horizontality condition (see [GO2]). But, in general, the action of G c on G/Cp does not preserve the horizontality condition.…”