Generalized Weierstrass representation for surfaces in Heisenberg spaces (Q763048)

It is well-known that constant mean curvature surfaces immersed in Euclidean \(3\)-space admit an integral representation in terms of their Gauss maps which resembles the classical Weierstrass representation for minimal surfaces, and these also have harmonic Gauss map with target given by some Grassmannian space. This approach was applied to diverse different situations including \(3\)-dimensional Lie groups and \(3\)-dimensional homogeneous spaces by many people. Motivating this, in this paper, the authors search for analogs of these results for space-like surfaces immersed in \(3\)-dimensional Lorentzian homogeneous space. In this paper, the authors deal with both Riemannian and Lorentzian structures in the Heisenberg group in order to explore analogs and differences between two settings. They give a representation of a minimal immersion in terms of integrals involving a harmonic Gauss map whose target is a disk endowed either with the hyperbolic metric in the Riemannian case or with the spherical metric in the Lorentzian case. The harmonicity of the Gauss map and an integral representation of minimal surface in Riemannian Heisenberg groups were previously obtained in [\textit{B. Daniel}, Int. Math. Res. Not. 2011, No. 3, 674--695 (2011; Zbl 1209.53048)] and [\textit{F. Mercuri, S. Montaldo} and \textit{P. Piu}, Acta Math. Sin., Engl. Ser. 22, No. 6, 1603--1612 (2006; Zbl 1119.53041)].