Constant Mean Curvature Spacelike Surfaces in Lorentz-Minkowski Space

Abstract

In this chapter we focus on compact spacelike surfaces in Lorentz-Minkowski space \(\mathbb{L}^{3}\) with constant mean curvature spanning a given boundary curve. Again we are interested in studying the influence of the geometry of the boundary on the shape of the surface. Although the mean curvature equation for a such surface is elliptic again, we shall present some differences with the Euclidean setting. For example, although a flux formula is derived and which is formally identical to the Euclidean space, however we do not obtain similar consequences. In contrast, and in the case that the boundary is a circle, we prove that umbilical surfaces are the only compact spacelike cmc surfaces spanning a circle. The chapter finishes by considering the Dirichlet problem of the mean curvature equation for spacelike graphs. Based on the properties of the rotational cmc surfaces in \(\mathbb{L}^{3}\), which provide good barriers for the necessary C 1-estimates, it will be proved that given a bounded domain \(\varOmega\subset \mathbb{R}^{2}\), the solvability of the Dirichlet problem in \(\mathbb{L}^{3}\) is ensured for arbitrary boundary data and values of H.