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Constant mean curvature surfaces and loop groups. (English) Zbl 0779.53004

This paper investigates the connections between constant mean curvature (cmc) surfaces in three dimensional space and solutions of the sinh- Gordon equation. The compatibility condition of the Frenet equations [U. Pinkall and I. Sterling, Ann. Math., II. Ser. 130, No. 2, 407-451 (1989; Zbl 0683.53053)] for the moving frame of a cmc surface is equivalent to the sinh-Gordon equation. Formally, the Frenet equations can be obtained from a loop group via a Riemann-Hilbert splitting. This allows the authors to show that the space of solutions to the sinh-Gordon equation (and hence the corresponding cmc surfaces) form a Banach manifold and that certain “Jacobi” vector fields discussed by Pinkall and Sterling [loc. cit.] correspond to a hierarchy of equations associated to this splitting. Finally it is shown that all finite type solutions are contained in this Banach manifold.
Reviewer: R.Schmid (Atlanta)

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
20N05 Loops, quasigroups
58C50 Analysis on supermanifolds or graded manifolds

Citations:

Zbl 0683.53053
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