Geometry of a class of $\phi$-minimal surfaces in ${\mathbb{H}}^2\times \mathbb{R}$

We study a class of $\phi$-minimal surfaces in the product space ${\mathbb{H}}^2\times \mathbb{R}$, that is, weighted minimal surfaces whose corresponding weight is determined by a smooth potential function $\phi$ that depends only on the height coordinate relative to the factor $\mathbb{R}$. Under monotonicity, concavity, and asymptotic conditions on $\phi$, we obtain a complete classification of connected, properly embedded $\phi$-minimal surfaces that are invariant under isometries of ${\mathbb{H}}^2\times \mathbb{R}$ acting trivially on $\mathbb{R}$ and by rotations or parabolic translations on ${\mathbb{H}}^2$. Finally, we establish several non-existence results for $\phi$-minimal surfaces in ${\mathbb{H}}^2\times \mathbb{R}$, either by using properties of the height function or by applying the tangency principle together with suitable invariant surfaces as barriers.