The Alexandrov problem in a quotient space of $\mathbb{H}^2 \times \mathbb{R}$

In this talk, we will prove an Alexandrov type theorem for a quotient space of $\mathbb{H}^2 \times \mathbb{R}$. More precisely, we will 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 parabolic translation along horocycles of $\mathbb{H}^2$ and a vertical translation. Moreover, we will construct some examples of periodic minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ .