Area bound for surfaces in generic gravitational field

We define an attractive gravity probe surface (AGPS) as a compact 2-surface $S_alpha$ with positive mean curvature $k$ satisfying $r^a D_a k / k^2 ge alpha$ (for a constant $alpha>-1/2$) in the local inverse mean curvature flow, where $r^a D_a k$ is the derivative of $k$ in the outward unit normal direction. For asymptotically flat spaces, any AGPS is proved to satisfy the areal inequality $A_alpha le 4pi [ ( 3+4alpha)/(1+2alpha) ]^2(Gm)^2$, where $A_{alpha}$ is the area of $S_alpha$ and $m$ is the Arnowitt-Deser-Misner (ADM) mass. Equality is realized when the space is isometric to the $t=$constant hypersurface of the Schwarzschild spacetime and $S_alpha$ is an $r=mathrm{constant}$ surface with $r^a D_a k / k^2 = alpha$. We adapt the two methods of the inverse mean curvature flow and the conformal flow. Therefore, our result is applicable to the case where $S_alpha$ has multiple components. For anti-de Sitter (AdS) spaces, a similar inequality is derived, but the proof is performed only by using the inverse mean curvature flow. We also discuss the cases with asymptotically locally AdS spaces.