2023/06/21 by Del Prete, Andrea, Manzano, José M., Nelli, Barbara
#53A10 (Primary) #53C30 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.12195
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.