Investigation and Application of the Dressing Action on Surfaces of Constant Mean Curvature

Abstract

The dressing method for the generating of new solutions of integrable systems was introduced in 1979 by Zakharov and Shabat [16]. It was soon traced back to a natural loop group action on the solution space of integrable systems [1,11]. With the introduction of the theory of integrable systems into geometry, most notably the theory of surfaces of constant mean curvature (CMC surfaces) [10,2], the dressing method also entered the realm of differential geometry.Let it serve as an illustration of the power of the dressing action that, due to [8] (see also [3]), all CMC immersions of finite type, in particular all CMC tori, are included in the dressing orbit of the standard cylinder. This result was used in [4] to reproduce the classification of CMC tori given in [10] in terms of the dressing action.Moreover, while the integrable systems methods in general only apply to a certain class of CMC surfaces, those without umbilics, the dressing action can easily be applied also to CMC surfaces with umbilics. The latter was made possible by a general loop group theoretic approach to CMC surfaces, the so called DPW method [7].Similar to the Weierstraß representation of minimal surfaces, the DPW method starts with a holomorphic function E and a meromorphic function f , both defined on some open, simply connected subset D of the complex plane, and constructs an S 1 -family of isometric conformal CMC immersions Ψ λ∈S 1 : D → R 3 , the associated family, from these data. As a first application of the dressing action, in [6] and [15] it was shown that the dressing action can be used to describe the set of admissible input data (E, f ) for the DPW method.While Edz 2 is simply the Hopf differential of the resulting CMC immersion, the function f has no such simple geometric interpretation. The problem shows especially if one wants to construct CMC immersions Ψ which are invariant under a symmetry group Γ ⊂ AutD of biholomorphic automorphisms of D, i.e. Ψ • γ = T Ψ, where γ ∈ Γ and T is a (proper) Euclidean motion in space. It is clear, that Edz 2 has to be automorphic w.r.t. Γ, i.e. γ * (Edz 2 ) = Edz 2 or E • γ = (γ ′ ) −2 E. But the meromorphic function f satisfies no such automorphicity conditions. Instead, as was shown by the authors in [5], f transforms by complicated dressing transformations under Γ. For further reference see [5]. Thus, in the DPW formulation the understanding of compact and symmetric CMC immersions is intimately related to the understanding of the dressing action. This served as a strong incentive to further investigate the dressing action, in particular on the meromorphic data of the DPW method.Clearly, the Hopf differential is invariant under dressing. Wu [14] constructed a large set of algebraic invariants under dressing and was able to find normalized representatives in each dressing orbit [13]. These results are particularly useful, since the dressing orbits, as the finite type results indicate, are very large.

Harmonic maps of finite uniton type into inner symmetric spaces

“…Recall (e.g. [12], see also [17], [3], [11], [26]) that for h + ∈ Λ + G C σ and the extended frame F of some harmonic map F : M → G/K we consider the Iwasawa splitting…”

Bubbletons are not embedded

“…Recently the classical transformations have received a treatment from the modern point of view of dressing [2,3,4,5,6,7,8,10,11,12,19]. Bubbletons can be realized by dressing the round cylinder by a class of very simple maps, called simple factors [19].…”

Ribaucour Transformations for Hypersurfaces in Space Forms

“…The following methods were used in the construction of cmc surfaces: the method of perturbation [10,11]; integrable systems [2,15]; conjugate prime surfaces [12,14]; and Weierstrass type representation [8,13]. Recently, complete cmc surfaces were constructed by applying another method, based on Ribaucour transformations.…”