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Surfaces in terms of 2 by 2 matrices. Old and new integrable cases. (English) Zbl 0841.53003
Fordy, Allan P. (ed.) et al., Harmonic maps and integrable systems. Based on conference, held at Leeds, GB, May 1992. Braunschweig: Vieweg. Aspects Math. E23, 83-127 (1994).

Families of surfaces on which some geometric characteristic remain unchanged have been a traditional object of study in classical differential geometry. Examples include families of isometric surfaces, families of surfaces with constant mean or Gauss curvature, and many others. Many beautiful geometric results were discovered in the past by classical geometers and continue to be discovered (and rediscovered) presently. A new point of view at the subject, connected with the much younger soliton theory, provided an extremely useful framework in which such families of surfaces can be identified with families of solutions to associated nonlinear equations, for which new solutions can be generated from the ones that are already known. Furthermore, the procedure(s) for this process can be described analytically and effectively, and new solutions (surfaces) can be numerically computed.

In this article the author surveys eight families of surfaces (most of them were known to classical geometers) and describes in detail the corresponding integrable equations from the point of view of soliton theory. The survey contains many results, new as well as already known. It is very well written and shows clearly the connections between geometric and analytic properties of solutions.

53A05Surfaces in Euclidean space
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
58E12Applications of variational methods to minimal surfaces
35Q51Soliton-like equations