Minimal surfaces on three-dimensional Walker manifolds

Abstract

Lorentzian Geometry has shown to be very useful in a wide range of studies including many diverse research elds, especially in the theory of general relativity and mathematical cosmology. A Walker manifold descends from the structure of Lorentzian manifolds which is characterized by admitting a parallel degenerate distribution. In the present thesis, we investigate and derive the equations of minimal surfaces on three-dimensional Walker manifolds, with a particular interest on those surfaces which are represented by the graph of a smooth function. Our study is closely related with (Lorentzian) isothermal coordinates which provide an easier approach for deriving such equations, and they are locally de ned for any surface on the underlying manifold. By using the well-known property of vanishing mean curvature for minimal surfaces, together with the geometric restrictions posed by the chosen coordinates, we obtain a class of graphs of functions which are minimal under certain conditions on the corresponding function.