In this work we present some results on minimal and constant mean curvature surfaces in homogeneous $3$-manifolds.
First, we classify the compact embedded surfaces with constant mean curvature in the quotient of $\mathbb H^2\times\mathbb R$ by a subgroup of isometries generated by a horizontal translation along horocycles of $\mathbb H^2$ and a vertical translation. Moreover, in $\mathbb H^2\times\mathbb R$, we construct new examples of periodic minimal surfaces and we prove a multi-valued Rado theorem for small perturbations of the helicoid.
In some metric semidirect products, we construct new examples of complete minimal surfaces similar to the doubly and singly periodic Scherk minimal surfaces in $\mathbb R^3.$ In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in $Sol_3$ with a one-parameter family of non-isometric metrics.
After that, we prove a half-space theorem for an ideal Scherk graph $\Sigma\subset M\times\mathbb R$ over a polygonal domain $D\subset M$, where $M$ is a Hadamard surface with bounded curvature. More precisely, we show that a properly immersed minimal surface contained in $D\times\mathbb R$ and disjoint from $\Sigma$ is a translate of $\Sigma.$
Finally, based in a joint paper with L. Hauswirth, we prove that if a properly immersed minimal surface in the quotient space $\mathbb H^2\times\mathbb R/G$ has finite total curvature then its total curvature is a multiple of $2\pi,$ and moreover, we understand the geometry of the ends. Here $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb H^2$ without fixed points.