  Let $V$ be a maximal globally hyperbolic flat $n+1$--dimensional space--time
with compact Cauchy surface of hyperbolic type. We prove that $V$ is globally
foliated by constant mean curvature hypersurfaces $M_{\tau}$, with mean
curvature $\tau$ taking all values in $(-\infty, 0)$. For $n \geq 3$, define
the rescaled volume of $M_{\tau}$ by $\Ham = |\tau|^n \Vol(M,g)$, where $g$ is
the induced metric. Then $\Ham \geq n^n \Vol(M,g_0)$ where $g_0$ is the
hyperbolic metric on $M$ with sectional curvature -1. Equality holds if and
only if $(M,g)$ is isometric to $(M,g_0)$.
