Minimal graphs in three-dimensional Killing submersions

We consider a connected and orientable Riemannian (resp. Lorentzian) three-manifold E admitting a never vanishing (resp. temporal) complete Killing vector field ξ ∈ X(E) whose associated one-parameter group of isometries G of E acts freely and properly on E. Then, there exists a Killing submersion π: E → M = E/G whose fibers are the integral curves of ξ. Killing submersions give rise to a natural notion of graph over a domain in M, that is, a smooth section of π over this domain. In this setting 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. 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 solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of M, taking piecewise continuous boundary values. We study the uniqueness of solutions of the Dirichlet problem over unbounded domains of M obtaining a general Collin-Krust type estimate. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. Finally, we develop a conformal duality for spacelike graphs in Riemannian and Lorentzian Riemannian and Lorentzian Killing submersions. The duality swaps mean curvature and bundle curvature and sends the length of the Killing vector field to its reciprocal while keeping invariant the base surface. We obtain two consequences of this result. On the one hand, we find entire graphs in Lorentz–Minkowski space L3 with prescribed mean curvature a bounded function H∈C∞(R2) with bounded gradient. On the other hand, we obtain conditions for existence and non existence of entire graphs which are related to a notion of critical mean curvature.