Lemma 4.4: smooth compact comparison region

Choose the supplied section at this fixed $q>0$ from the maximal-slice family, taking any lower parameter bound strictly between zero and $q$. Thus the same section is a complete smooth spacelike maximal stationary graph, as required by the ADM theorem. The original maximal-section, mass-one, regular-vacuum and $T{\mathrm{\mid }}_S\equiv ̸0$ hypotheses are exactly the standing hypotheses, and ZFC supplies its choice convention. Apply [F8] to obtain temporal $v$ and $N>0$ in the interior. The metric and section are smooth up to $S$ by [F6] and [F7], so their normal, lapse and shift extend smoothly there. Regularity makes $T$ tangent to both transversely intersecting horizon branches, hence to their intersection $S$. Since $TS\subset T\mathrm{\Sigma }$ is orthogonal to $n$, $N=-g_q(T,n)=0$ on $S$ and $X{\mathrm{\mid }}_S=T{\mathrm{\mid }}_S\in TS$. Stationarity of $g_q$ and the section-time conventions provide the stationary extensions to the translated section interiors.

The given sufficiently small smooth distance collar is a compact boundary-distance collar in this smooth stationary geometry, with $\rho$ equal to intrinsic distance and a slightly larger smooth collar available. Step 1.1 supplies $N>0$ inside, $N=0$ at $S$, tangency of $T$ to $S$, and stationary extensions. The section-time product and $Tv=1$ are those of [F8]. Thus every hypothesis of [F9] holds. Apply it to obtain a common smooth normal flow on $\mathcal{R}$, fixing $S$ and with $v({\mathrm{\Psi }}_{z}x)=z$ for interior $x$, for all sufficiently small $\mathrm{\mid }z\mathrm{\mid }$. Restrict its common interval to $\mathrm{\mid }z\mathrm{\mid }\le 2\delta$ for all sufficiently small $\delta >0$. Its interior trajectories stay in the exterior. This supplies the flow and its boundary extension required by the truncated-height definition without assuming them from the height theorem.

The fixed positive $q$, original maximal exterior section, $T{\mathrm{\mid }}_S\equiv ̸0$, supplied complete smooth maximal section and sufficiently small distance collar are the standing data of localization, the endpoint theorem and the height result. The sufficiently late target is chosen to meet their late-target thresholds before it is fixed, independently of $ϵ$. Localization supplies the compact zero-time cut $D$ in the collar. Apply the endpoint theorem to that $D$ and the same $C$ to obtain its maximal past null-normal family, compact Borel $A\subset C$ and endpoint map $F:A\to D$ with local smooth extensions.

For these same section, collar and target data, temporal $v$ from step 1.1 and the common boundary-fixing normal flow from step 2.1 meet the inputs of the truncated-height definition. With the sufficiently late target fixed independently of $ϵ$, apply the height result [F5] for a suitable sufficiently small $\delta >0$ in the common flow interval. It gives the truncated supremum height $u$, Lipschitz with the two constant boundary neighborhoods, and identifies its zero-time endpoint data with those in step 2.2. This supplies the height and maximal-family endpoint construction for [F2], whose own conclusion selects a suitable height parameter for each $ϵ$.

Apply the coarea-smoothing result with the given $ϵ$ to the fixed section, compact smooth collar and late target, its induced metric $h_0$, the height construction in step 3.1 and the endpoint data in step 2.2. Its own interface chooses a sufficiently small height parameter for this $ϵ$. It gives a smooth compact sublevel region ${\mathrm{\Omega }}_{ϵ}\subset \mathcal{R}$ and regular interior level $D_{ϵ}$, with boundary $S\bigsqcup D_{ϵ}$ and $\mathrm{\mid }{D}_{ϵ}{\mathrm{\mid }}_{h_0}\le \mathrm{\mid }C\mathrm{\mid }+ϵ$.

Since $\mathcal{R}\subset \mathrm{\Sigma }$ and $D_{ϵ}$ is an interior level separating its boundary components, $D_{ϵ}$ lies between $S$ and $S_{\eta }$; the same supplier gives smoothness, closedness and possible disconnectedness. The induced $h_0$ area on $D_{ϵ}$ is its fixed-$q$ spacetime-induced area, while $\mathrm{\mid }C\mathrm{\mid }$ uses the induced metric on $C$. Thus the inequality is exactly the asserted one.