Compact λ-translators in H²× R with boundary

A λ λ -translator in H²× R H 2 × R is a surface whose mean curvature H satisfies H= N,∂ z +λ H = ⟨ N , ∂ z ⟩ + λ , where N is the unit normal of the surface, ∂ z ∂ z is the vertical Killing vector field and λ ∈ R λ ∈ R . In this paper, we study how the geometry of the boundary of a compact λ 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 λ -translator is a graph. Finally, we present estimates for the area of a vertical graph λ λ -translator in terms of its height and volume.