Compact $$\lambda $$-translators in $$\mathbb {H}^2\times \mathbb {R}$$ with boundary

Abstract A $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -translator in $$\mathbb {H}^2\times \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> is a surface whose mean curvature H satisfies $$H= \langle N,\partial _z\rangle +\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>z</mml:mi> </mml:msub> <mml:mo>⟩</mml:mo> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> </mml:mrow> </mml:math> , where N is the unit normal of the surface, $$\partial _z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>z</mml:mi> </mml:msub> </mml:math> is the vertical Killing vector field and $$\lambda \in \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . In this paper, we study how the geometry of the boundary of a compact $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -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 $$\mathbb {H}^2\times \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> and the geometry of $$\mathbb {H}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , 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 $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -translator is a graph. Finally, we present estimates for the area of a vertical graph $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -translator in terms of its height and volume.