On a Liu–Yau type inequality for surfaces
Résumé
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.
Origine : Fichiers produits par l'(les) auteur(s)
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