The stability of a minimal surface in a pseudo-Euclidean space

We prove that any piece of a space-like minimal surface of the form z=z(x, y), in the pseudo-Euclidean space E^3,1 with line element dx²+dy²–dz² has the greatest area among close space-like surfaces with the same boundary. For time-like surfaces the following assertion is proved. Let V² be a time-like minimal surface in E^3,1. Then for any domain on V², there exists a smooth variation in the class of time-like surfaces with fixed boundary that increases the area, and there exists a smooth variation in the same class of surfaces that decreases the area.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 41–45, 1990.