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On the dressing action of loop groups on constant mean curvature surfaces. (English) Zbl 0912.53010

Surfaces of constant mean curvature in space can be constructed via a generalized Weierstrass representation from a holomorphic (quadratic) differential, the Hopf differential, and a meromorphic one-form, which is closely related with the first fundamental form, but does not seem to have a direct geometric meaning. Even though this construction produces all surfaces of constant mean curvature with \(H\neq 0\) in a unique way, other means of constructing such surfaces are very useful in many cases. One of these is a deformation technique, known from soliton theory as “dressing”. As a matter of fact, an infinite-dimensional group acts via dressing on the set of surfaces of constant mean curvature. One can show, e.g., that all tori can be obtained from the standard cylinder via dressing. This leads naturally to the question of characterizing the dressing orbits. This problem is closely related with finding all dressing invariants. It is easy to see that the Hopf differential is such an invariant.
The paper under review discusses dressing invariants of the type \(f(p'(0), p''(0),\dots, p^{(k)}(0))\), where 0 is a fixed base point in the surface and \(p\) is the coefficient of the meromorphic one form in the Weierstrass data for the surface, and \(k\) is an integer. It is shown that there is no dressing invariant if 0 is not an umbilic and the dressing invariants are determined if 0 is a umbilic.
As a consequence it is shown that the dressing orbits are not parametrized by the Hopf differential, and an interesting example, illustrating this, is given. The end of the paper provides an explicit recursive construction of all dressing transformations (as elements of a loop group). In addition, it is shown among other things that, if 0 is an umbilic of order at least 2, the Fourier coefficients of a possible dressing transformation, deforming the surface into another, exist and are uniquely determined if and only if all dressing invariants determined in this paper take the same values for the Weierstrass data of the two surfaces. An actual dressing transformation does exist iff the corresponding Fourier series converges.
This is a well-written paper. It marks a major step forward towards a classification of all dressing orbits and thus to a better understanding of the class of surfaces of constant mean curvature.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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