The Jenkins-Serrin problem in 3-manifolds with a Killing vector field

We consider a Riemannian submersion from a 3-manifold E to a surface M , both connected and orientable, whose fibers are the integral curves of a Killing vector field without zeros, not necessarily unitary. We solve the Jenkins–Serrin problem for the minimal surface equation in E over a relatively compact open domain Ω ⊂M with prescribed finite or infinite values on some arcs of the boundary under the only assumption that the same value +∞or −∞cannot be prescribed on two adjacent components of ∂Ω forming a convex angle. The domain Ω can have reentrant corners as well as closed curves in its boundary. We show that the solution exists if and only if some generalized Jenkins–Serrin conditions (in terms of a conformal metric in M ) are fulfilled. We develop further the theory of divergence lines to study the convergence of a sequence of minimal graphs. We also provide maximum principles that guarantee the uniqueness of the solution. Finally, we obtain new examples of minimal surfaces in R3 and in other homogeneous 3-manifolds.