On a Liu--Yau type inequality for surfaces

Let $Ω$ be a compact and mean-convex domain with smooth boundary $Σ:=\partialΩ$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $Σ$ is spacelike in a spacetime $(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that $Σ$ admits an isometric and isospin immersion into $\mathbb{R}^3$ with mean curvature $H\_0$, then: \begin{eqnarray*} \int\_Σ|\mathcal{H}|dΣ\leq\int\_Σ\frac{H\_0^2}{|\mathcal{H}|}dΣ. \end{eqnarray*} If equality occurs, we prove that there exists a local isometric immersion of $Ω$ 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.

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