1.7. Isothermic surfaces 25 The theory of discrete isothermic surfaces unifies the theories of smooth isothermic surfaces and of their Darboux transformations. M¨ obius-geometric characterization of isothermic surfaces and their Darboux transformations. It is easily checked that conditions (1.71) are invariant with respect to Euclidean motions and dilations in RN , as well as with respect to the inversion f → f/ f, f . In other words, the notion of isothermic surfaces belongs to the M¨ obius differential geometry. The same holds for their Darboux transformations. Therefore, it is useful to charac- terize these notions within the M¨ obius-geometric formalism. (However, the notion of the dual surface, or Christoffel transformation, is essentially based on the Euclidean structure of the ambient space RN .) To find such a characterization (due to Darboux), note first of all that eqs. (1.72) are equivalent to ∂1∂2 ˆ = (∂2 log s)∂1 ˆ+ (∂1 log s)∂2 ˆ for the image ˆ : R2 → QN 0 of f in the quadric QN 0 ⊂ LN+1,1. Theorem 1.32. (Isothermic surfaces = Moutard nets in the light cone) The lift ˆ = s−1 ˆ : R2 → LN+1,1 of an isothermic surface f : R2 → RN to the light cone of RN+1,1 satisfies the Moutard equation (1.78) ∂1∂2ˆ = q12ˆ with q12 = s∂1∂2(s−1) and is the Moutard representative of f. Conversely, given a Moutard net ˆ : R2 → LN+1,1 in the light cone, define s : R2 → R∗ and f : R2 → RN by ˆ = s−1(f + e0 + |f|2e∞) (so that s−1 is the e0-component, and s−1f is the RN -part of ˆ in the basis e1, . . . , eN , e0, e∞). Then f is an isothermic surface, and the definition (1.71) holds with the functions αi = ∂iˆ ∂iˆ depending on ui only. Note that in the second part of the theorem we can always assume that s : R2 → R+, changing ˆ to −ˆ, if necessary. Thus, we see that the isothermic surfaces are in a one-to-one corre- spondence with the Moutard nets in LN+1,1, i.e., with Lorentz-harmonic LN+1,1-valued functions. Let us address the problem of minimal data which determine an isother- mic surface (i.e., a Moutard net in LN+1,1) uniquely. Guided by an analogy with the case of K-surfaces, one is tempted to think that two arbitrary curves ˆ Bi in LN+1,1 would be such data. However, as a consequence of the fact that now we are dealing with the light cone LN+1,1 = {ˆ ˆ = 0} rather than with the sphere S2 = {n, n = 1} as a quadric where Moutard nets

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