  Let $\Omega$ be a compact and mean-convex domain with smooth boundary
$\Sigma:=\partial\Omega$, in an initial data set $(M^3,g,K)$, which has no
apparent horizon in its interior. If $\Sigma$ is spacelike in a spacetime
$(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that
$\Sigma$ admits an isometric and isospin immersion into $\mathbb{R}^3$ with
mean curvature $H\_0$, then: \begin{eqnarray*}
\int\_{\Sigma}|\mathcal{H}|d\Sigma\leq\int\_{\Sigma}\frac{H\_0^2}{|\mathcal{H}|}d\Sigma.
\end{eqnarray*} If equality occurs, we prove that there exists a local
isometric immersion of $\Omega$ in $\mathbb{R}^{3,1}$ (the Minkowski spacetime)
with second fundamental form given by $K$. In Theorem liu-yau-minkowski, we
also examine, under weaker conditions, the case where the spacetime is the
$(n+2)$-dimensional Minkowski space $\mathbb{R}^{n+1,1}$ and establish a
stronger rigidity result.
