Surfaces in three-dimensional Lie groups

We derive the Weierstrass (or spinor) representation for surfaces in the three-dimensional Lie groups Nil,$\widetilde{SL}_2$ , and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. Using the spectral properties of the corresponding Dirac operators, we find analogs of the Willmore functional for surfaces in these geometries.