Minimal graphs in three-dimensional Killing submersions

Abstract

The goal of this thesis is to enrich the theory of minimal graphs in three-dimensional Killing submersions. A Killing submersion is a Riemannian submersion from a three-dimensional manifold E onto a Riemannian surface M whose fibers are integral curves of a Killing field. In this context, a Killing graph is a smooth section of the submersion. In this thesis, we study three problems. First, we solve the Jenkins-Serrin problem for the minimal surface equation over relatively compact domains of M with prescribed (possibly infinite) boundary values. Second, we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of M, taking piecewise continuous boundary values, and study the uniqueness of solutions over unbounded domains of M obtaining general Collin-Krust type estimates. Finally, we develop a conformal duality for spacelike graphs in Riemannian and Lorentzian Killing submersions with applications to the existence of entire graphs with prescribed mean curvature.