Surfaces in the four-space and the Davey–Stewartson equations

The Weierstrass representation for surfaces in R3 [8, 13] was generalized for surfaces in R4 in [12] (see also [4]). This paper uses the quaternion language and the explicit formulas for such a representation were written by Konopelchenko in [9] for constructing surfaces which admit soliton deformation governed by the Davey–Stewartson equations. This generalizes his results from [8] where he introduced the formulas for inducing surfaces in the three-space which involve a Dirac type equations and defined for such surfaces a deformation governed by the modified Novikov–Veselov (mNV) equations. It was shown in [13] that the formulas for inducing surfaces in R3 [8] describe all surfaces and that the modified Novikov–Veselov equation deforms tori into tori preserving the Willmore functional which naturally arises and plays an important role in this representation. The spectral curve of the corresponding Dirac operator is invariant under this deformation. The global Weierstrass representation at least for real analytic surfaces could be obtained by an analytic continuation from a local representation. Thus a moduli space of immersed tori is embedded into the phase space of an integrable system with the Willmore functional and, moreover, the spectral curve as conservation quantities. Looking forward to understand the spectral curves for tori in R4 we consider in this paper the analogous problems for surfaces in R4 and show that this case is very different from the three-dimensional case, in particular, by the following features which were overlooked until recently: