Area monotonicity for spacelike surfaces with constant mean curvature (Q556224)

Compact space-like surfaces \(\Sigma\) with constant mean curvature \(H\) in the three-dimensional Lorentz-Minkowski space \(\mathbb{L}^3\) are considered, assuming that the boundary is a plane curve. An estimate is obtained for the height \(h\) of the surface measured from the plane \(\Pi\) that contains the boundary. It is established that \(h\leq A|H|/(2\pi)\), where \(A\) is the area of the surface that lies over \(\Pi\). Equality holds if and only if \(\Sigma\) is a planar domain \((H= 0)\) or a hyperbolic cap \((H\neq 0)\) (i.e., a compact piece of a hyperbolic plane with the circular boundary.