In this talk we will present a version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol3. Moreover, we will construct examples of admissible domains where the results may be applied. This is a joint work with Patricia Klaser.

In this talk we will consider two uniqueness problems in \(\mathbb{H}^2\times\mathbb{R}\). First, we will prove a halfspace theorem for an ideal Scherk graph \(S\) over a polygonal domain $D$ in \(\mathbb{H}^2\), that is, we will show that a properly immersed minimal surface contained in \(D\times\mathbb{R}\) and disjoint from \(S\) is a translate of \(S\). Second, we will consider a multi-valued Rado theorem for small perturbations of the Helicoid. More precisely, we will prove that for certain small perturbations of the boundary of a (compact) helicoid there exists only one minimal disk with that boundary.