Complex analysis,maximal immersions and metric singularities

This paper introduces a complex analysis for the wave equation and for a singular second-order partial differential equation. As a main application of this complex analysis we construct type changing zero mean curvature immersions into Minkowski space. We also prove the existence of isothermal coordinates on a Lorentzian surface using this complex analysis and characterize flat maximal surfaces by their Gauss image. Finally we study the metric singularities of maximal immersions and semi Riemannian manifolds in general.