26 1. Classical Differential Geometry live, we cannot find an expression for q12 in terms of the first derivatives of ˆ anymore instead, one has: q12 = ∂3ˆ 1 ∂2ˆ ∂1ˆ ∂1ˆ = ∂3ˆ 2 ∂1ˆ ∂2ˆ ∂2ˆ . This shows that the coordinate curves ˆ B i are not arbitrary but rather subject to certain further conditions. We leave the question of correct initial data for an isothermic surface open. Darboux pairs of isothermic surfaces are characterized in terms of their Moutard representatives as follows. Theorem 1.33. (Darboux transformation = Moutard transforma- tion in the light cone) The lifts ˆ ˆ+ : R2 LN+1,1 of a Darboux pair of isothermic surfaces f, f + : R2 RN are related by a Moutard transfor- mation, i.e., there exist two functions p1, p2 : R2 R such that (1.79) ∂1ˆ+ + ∂1ˆ = p1(ˆ+ ˆ), ∂2ˆ+ ∂2ˆ = p2(ˆ+ + ˆ). Conversely, for a Moutard net ˆ in the light cone LN+1,1, any Moutard transform ˆ+ with values in LN+1,1 is a lift of a Darboux transform f + of the isothermic surface f. Note that the quantity ˆ ˆ+ is constant (does not depend on u R2), and is related to the parameter c of the Darboux transformation: ˆ ˆ+ = −c/2. The formulas (1.80) pi = ˆ ∂iˆ+− ∂iˆ ˆ+ 2 ˆ ˆ+ = ∂iˆ ˆ+ ˆ ˆ+ , i = 1, 2, make it apparent that a Moutard transform ˆ+ is completely determined by prescribing its value ˆ+(0) at one point. Indeed, eqs. (1.79) with coef- ficients (1.80) form a compatible system of first order differential equations for ˆ+ : R2 LN+1,1. Of course, the data (D1,2) are encoded in ˆ+(0) in a straightforward manner. 1.8. Surfaces with constant mean curvature Definition 1.34. (Surface with constant curvature) A surface f : R2 R3 is called a surface with constant mean (resp. Gaussian) curva- ture if the function H (resp. the function K) on the surface is constant. Surfaces with the identically vanishing mean curvature function are called minimal. Surfaces with constant curvature possess remarkable geometric proper- ties.
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