Some results on g-stability for hypersurfaces in an initial data set

We study the $g$-stability of hypersurfaces ${\Sigma }^{n-1}$ with null expansion ${\theta }^{+}=h\ge 0$ in an $n$-dimensional initial data set $M^n$ with cosmological constant $\Lambda$. First, under natural energy conditions, we demonstrate that ${\Sigma }^{n-1}\subset M^n$ admits a metric with positive scalar curvature. Second, for a $g$-stable surface ${\Sigma }^2$ of genus $g\left(\Sigma \right)$, we establish an inequality relating the area of $\Sigma$, its genus, $\Lambda$, and the charge $q\left(\Sigma \right)$. Moreover, if equality holds and $\Lambda >0$, ${\Sigma }^2$ is a round 2-sphere. Finally, for a $g$-stable, two-sided, closed hypersurface ${\Sigma }^4$ in a 5-dimensional initial data set $M^5$ satisfying natural energy conditions, we derive an inequality involving the area of $\Sigma$, its charge $q\left(\Sigma \right)$, and a positive constant depending on the total traceless Ricci curvature of $\Sigma$. Equality implies that ${\Sigma }^4$ is isometric to $S^4$.