Stationary bands in three-dimensional Minkowski space

Abstract

In this paper we consider a free boundary problem for spacelike surfaces in the 3-dimensional Lorentz-Minkowski space $\mathbb{L}^{3}$ whose energy functional involves the area of a surface and a timelike potential. The critical points of this energy for any volume-preserving admissible variation are spacelike surfaces supported in a plane and whose mean curvature is a linear function of the time coordinate. In this paper, we consider those surfaces that are invariant in a parallel coordinate to the support plane. We call these surfaces stationary bands. We establish existence of such surfaces and we investigate their qualitative properties. Finally, we give estimates of its size in terms of the initial data.