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

Abstract

We prove that any piece of a space-like minimal surface of the form z=z(x, y), in the pseudo-Euclidean space E3,1 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 E3,1. 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.