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We consider a Riemannian submersion from a 3-manifold $\mathbb{E}$ to asurface $M$, both connected and orientable, whose fibers are the integralcurves of a Killing vector field without zeros, not necessarily unitary. Wesolve the Jenkins-Serrin problem for the minimal surface equation in$\mathbb{E}$ over a relatively compact open domain $\Omega\subset M$ withprescribed finite or infinite values on some arcs of the boundary under theonly assumption that the same value $+\infty$ or $-\infty$ cannot be prescribedon two adjacent components of $\partial\Omega$ forming a convex angle. Thedomain $\Omega$ can have reentrant corners as well as closed curves in itsboundary. We show that the solution exists if and only if some generalizedJenkins-Serrin conditions (in terms of a conformal metric in $M$) arefulfilled. We develop further the theory of divergence lines to study theconvergence of a sequence of minimal graphs. We also provide maximum principlesthat guarantee the uniqueness of the solution. Finally, we obtain new examplesof minimal surfaces in $\mathbb{R}^3$ and in other homogeneous $3$-manifolds.

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