Weierstrass Representations for Surfaces in 4D Spaces and Their Integrable Deformations via DS Hierarchy

Generalized Weierstrass representations for generic surfaces conformally immersed into four-dimensional Euclidean and pseudo-Euclidean spaces of different signatures are presented. Integrable deformations of surfaces in these spaces generated by the Davey–Stewartson hierarchy of integrable equations are proposed. The Willmore functional of a surface is invariant under such deformations.