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Minimal surfaces in 3-dimensional solvable Lie groups II

This two-parameter family of solvable Lie groups contains the following particular examples: Euclidean 3-space E, hyperbolic 3-space H and Euclidean motion group £(1,1). Moreover, G(0,1) is isometric to the Riemannian direct product H x E of hyperbolic 2-space and the real line E. In this paper, we investigate the normal Gauss maps for surfaces in G(/ii,/i2) and reformulate the integral representation formula of [3] in terms of the normal Gauss map. On the other hand, study of minimal surfaces in the reducible Riemannian symmetric space H x E has been started very recently by Rosenberg and his collaborators. See [9, 10]. In a recent paper [7], Mercuri, Montaldo and Piu obtained an integral representation formula for minimal surfaces in H x E ([7, Theorem 5.1]). Their formula coincides with our formula for G(0,1). Thus our formula is a unification of Goes-Simoes-Kokubu formula [2, 5] and Mercuri-Montaldo-Piu formula [7].