M ar 2 00 5 Surfaces in three - dimensional Lie groups ∗

In the present paper we extend the methods of the Weierstrass (or spinor) representation of surfaces in R [10, 11] and SU(2) = S [12] for surfaces in the threedimensional Lie groups Nil , S̃L2, and Sol endowed with so-called Thurston’s geometries [9]. The main feature of this approach is that the geometry of a surface is related to the spectral properties of the corresponding Dirac operator. Therewith this approach reveals some unknown before geometric meanings of the Willmore functional and the Willmore conjecture which states that for tori the Willmore functional attains its minimum on the Clifford torus. A function ψ generates a surface in R via the Weierstrass formulas if and only if it meets some equation of the Dirac type where the Dirac operator has, in general, two potentials U and V which coincide for the case of surfaces in R. Given a surface M in R, the integral