9 Area minimization among marginally trapped surfaces in Lorentz - Minkowski space

We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values. The quintessential property of zero mean curvature surfaces in Euclidean space is that they locally minimize area with respect to their boundary curves. The method of calibrations can be used to show that any part of such a surface which can be represented as a graph over a convex domain, has least area when compared with any surface sharing the same boundary values. This justifies calling surfaces with zero mean curvature surfaces in Euclidean space 'minimal surfaces'. In a similar way, zero mean curvature space-like surfaces in three dimensional Lorentz-Minkowski space are well known to locally maximize area. For space-like zero mean curvature surfaces in any four dimensional Lorentzian manifold there can be no analogous local minimizing or local maximizing property. An arbitrarily small neighborhood of any point possesses an infinite dimensional space of deformations of the surface which fix the boundary and decrease the area and it has an infinite dimensional space of deformations which fix the boundary and increase the area. In a sufficiently small neighborhood of a point, any compactly supported variation with space-like,