Let \({\mathbb {E}}\) be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field whose associated one-parameter group of isometries of \({\mathbb {E}}\) acts freely and properly on \({\mathbb {E}}\) . Then, there exists a Killing submersion from \({\mathbb {E}}\) 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.
