Minimal surfaces in Lorentzian Heisenberg group and Damek-Ricci spaces via the Weierstrass representation

In this paper we discuss a Weierstrass type representation for minimal surfaces in the3-dimensional Heisenberg group and in the 4-dimensional Damek-Ricci spaces, endowed with left invariant Riemannian or Lorentzian metrics. For the case of spacelike surfaces we employ the complex analysis, and for timelike surfaces our approach makes use of the paracomplex analysis. Then, we exhibit various examples of spacelike and timelike minimal surfaces in these spaces.