Flat maximal immersions
The author studies Lorentzian immersed surfaces with zero mean curvature in the three-dimensional Minkowski space \(\mathbb{R}^3_1\). Paralleling the classical terminology of minimal surfaces in Euclidean space, the author refers to this kind of surfaces as maximal surfaces. First, it is shown that a geodesically complete maximal surface in \(\mathbb{R}^3_1\) is flat if and only if it is the graph over a timelike plane of a function of one variable. On the other hand, it is well-known that zero mean curvature surfaces arise naturally as critical points of the area functional. In this respect, the author considers the stability of maximal Lorentzian surfaces in \(\mathbb{R}^3_1\), and he shows that they are always unstable, so that these surfaces neither locally maximize nor locally minimize the area. Finally, the author studies maximal Lorentzian surfaces of revolution. To the reviewer, the fact that Lorentzian surfaces with zero mean curvature in Minkowski 3-space are always unstable means that it does not seem very appropriate to refer to them as maximal or minimal surfaces. Maybe they should just be called stationary or critical surfaces.
scientific article; zbMATH DE number 866112