Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4‐d Spaces and Deformations via Ds Hierarchy

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.