Minimal Graphs in Riemannian Fibrations

In this dissertation, minimal and constant mean curvature surface theory in 3-dimensional Riemannian fibrations are studied. In the main part of the thesis, new complete, embedded minimal surfaces in the 3-sphere are constructed by solving a Plateau problem with respect to a suitable Jordan curve consisting entirely of horizontal geodesic arcs and extending this solution by means of Schwarz reflection. Additionally, an elementary proof for the vertical half-space theorem in Heisenberg space is given by finding a subsolution of the minimal surface equation. Finally, projections of constant mean curvature multigraphs are characterized: they are locally contained to one side of complete curves with constant geodesic curvature