If at a point z 0 ∈ D, f has a zero and g is holomorphic, then, as was pointed out in [5] the map Φ 1 fails to be an immersion at z = z 0 (see also the appendix).If g(z) has a zero at z = z 0 , and f is defined and nonzero there, then one gets an umbilic at the image of z 0 in R 3 .This allows one to use the DPW method, to construct a CMC immersion with a prescribed distribution of umbilics in the domain D.

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We investigate CMC-surfaces with periodic metric in a dressing orbit of the cylinder. It is shown, that such surfaces are always of finite type. Using the periodicity conditions for the extended frame of a CMC-surface, we develop an alternative approach to the classification of CMC-tori given by Pinkall and Sterling.

A general group theoretic formulation of integrable systems is presented. The approach generalizes the discussion of the KdV equations of Segal and Wilson based on ideas of Sato. The starting point is the construction of commuting flows on the group via left multiplication with elements from an abelian subgroup. The initial data are then coded by elements, called abstract scattering data, in a certain coset space. The resulting equations of motion are then derived from a suitably formulated Maurer-Cartan equation (zero curvature condition) given an abstract Birkhoff factorization. The resulting equations of motion are of the Zakharov-Shabat type. In the case of flows periodic in x-space, the integrals of motion have a natural group theoretic interpretation. A first example is provided by the generalized nonlinear Schrödinger equation, first studied by Fordy and Kulish with integrals of motion which may be local or nonlocal. A suitable reduction gives the mKdV equations of Drinfeld and Sokolov. On the level of abstract scattering data the generalized Miura transformation from solutions of the mKdV equations to the KdV type equations is then just the canonical map from a coset space to a double coset space. This group theoretic approach is related to the algebraic geometric discussion of integrable systems via an affine map from the abelian group describing flows restricted to a suitable set of abstract scattering data, called algebraic geometric, onto a connected component of the Picard variety.

Abstract. We construct a Grassmannian-like formulation for the potential KP-hierarchy including additional "negative" flows. Our approach will generalize the notion of a τ -function to include negative flows. We compare the resulting hierarchy with results by Hirota, Satsuma and Bogoyavlenskii.

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.

Abstract. We give a new proof of a theorem by M.P. do Carmo and M. Dajczer on helicoidal surfaces of constant mean curvature.