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We study minimal graphs in the homogeneous Riemannian 3-manifold$\widetilde{PSL_2(\mathbb{R})}$ and we give examples of invariant surfaces. Wederive a gradient estimate for solutions of the minimal surface equation inthis space and develop the machinery necessary to prove a Jenkins-Serrin typetheorem for solutions defined over bounded domains of the hyperbolic plane.

Minimal Surfaces in $\widetilde{PSL_2(\mathbb{R})}$