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

Abstract

We study a class of $φ$-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 $φ$ that depends only on the height coordinate relative to the factor $\mathbb{R}$. Under monotonicity, concavity, and asymptotic conditions on $φ$, we obtain a complete classification of connected, properly embedded $φ$-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 $φ$-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.