Minimal Surfaces in $\widetilde{PS{L}_2(\mathbb{R})}$

We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{PS{L}_2(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and develop the machinery necessary to prove a Jenkins-Serrin type theorem for solutions defined over bounded domains of the hyperbolic plane.