Abstract A λ -translator in H² R H 2 × R is a surface whose mean curvature H satisfies H= N, ᵦ + H = ⟨ N, ∂ z ⟩ + λ, where N is the unit normal of the surface, ᵦ ∂ z is the vertical Killing vector field and R λ ∈ R. In this paper, we study how the geometry of the boundary of a compact λ -translator affects the shape of the surface, asking under what conditions the symmetries of the boundary are inherited by the whole surface. Due to the product structure of H² R H 2 × R and the geometry of H² H 2, we distinguish between different notions of graphs and reflections. We provide conditions on the boundary curve of the surface to ensure that an embedded compact λ -translator is a graph. Finally, we present estimates for the area of a vertical graph λ -translator in terms of its height and volume.
Bueno et al. (Sat,) studied this question.