Minimal graphs over non-compact domains in 3-manifolds with a Killing vector field

Abstract Let $${\mathbb {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field whose associated one-parameter group of isometries of $${\mathbb {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> acts freely and properly on $${\mathbb {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> . Then, there exists a Killing submersion from $${\mathbb {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> onto a connected and orientable surface M whose fibers are the integral curves of the Killing vector field. In this setting, assuming that M is non-compact and the fibers have infinite length, we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of M , prescribing piecewise continuous boundary values. We obtain general Collin-Krust type estimates. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable.