Bg. Konopelchenko et G. Landolfi, Induced surfaces and their integrable dynamics II. Generalized Weierstrassrepresentations in 4-D spaces and deformations via DS hierarchy, STUD APPL M, 104(2), 2000, pp. 129-169
Extensions of the generalized Weierstrass representation to generic surface
s in 4-D Euclidean and pseudo-Euclidean spaces are given. Geometric charact
eristics of surfaces are calculated. It is shown that integrable deformatio
ns 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 t
otal squared mean curvature) is the simplest of them. Various particular cl
asses of surfaces and their integrable deformations are considered.