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Induced surfaces and their integrable dynamics. II: Generalized Weierstrass representations in 4-D spaces and deformations via DS hierarchy. (English) Zbl 1002.37034
Part I, cf. the first author, ibid. 96, 9-51 (1996; Zbl 0869.58027).
Summary: Extensions of the generalized Weierstrass representation to generic surfaces in 4-D Euclidean and pseudo-Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey-Stewartson hierarchy. Geometrically, these deformations are characterized by the invariance of an infinite set of functionals over surface. The Willmore functional (the total squared mean curvature) is the simplest of them. Various particular classes of surfaces and their integrable deformations are considered.
 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature