Compactness of the future horismos under null completeness

Put $F:=I^{+}(S)$. It is open because it is the union of the open sets $I^{+}(s)$, $s\in S$. If $x\in F$ and $x\ll y$, then some $s\in S$ satisfies $s\ll x$; transitivity gives $s\ll y$, so $y\in F$. Thus $F$ is an open future set.

For each point of $S$, choose a neighborhood with the two smooth null normal branches supplied by [F9]. On each branch rescale by a positive smooth factor so that $g(\mathrm{\ell },N)=-1$; this is possible by [F20] and does not change the sign of the expansion by [F8]. Both normalized expansions are continuous and negative at the chosen point by the trapped-surface hypothesis. Shrink the neighborhood so that both are at most $-c_j$ for some $c_j>0$. Compactness of $S$ gives a finite subcover; the minimum of its finitely many positive constants gives $c>0$ such that every normalized future null normal has initial expansion ${\theta }_0\le -c$.

By [F1] and [F3], $J^{+}(S)$ is closed, and $F\subseteq J^{+}(S)$; hence $\bar{F}\subseteq J^{+}(S)$. Therefore $\mathrm{\partial }F=\bar{F}\setminus F\subseteq E^{+}(S)$. Conversely, take $p\in E^{+}(S)$ and choose $s\in S$ with $s\le p$. The timelike vector $N_p$ and the local exponential map give future timelike points $p_{ϵ}\to p$ by [F12], [F14] and [F20]. If $s=p$ then $s\ll p_{ϵ}$; if $s\mathrm{\ne }p$ then $s\le p\ll p_{ϵ}$ and [F15] gives $s\ll p_{ϵ}$. Thus $p_{ϵ}\in F$ and, since $p\mathrm{\notin }F$, $p\in \mathrm{\partial }F$. Hence $E^{+}(S)=\mathrm{\partial }F$.

Set $L:=2\mathrm{/}c$. For each normalized null normal at $s\in S$, future completeness [F21], local smooth dependence [F16], and smooth evaluation on the maximal geodesic domain [F12] give a smooth null normal geodesic congruence near that initial datum through $[0,L]$. Applying [F10] with ${\theta }_0\le -c$ and the null Ricci hypothesis gives a first focal parameter ${\lambda }_f\le 2\mathrm{/}\mathrm{\mid }{\theta }_0\mathrm{\mid }\le L$. The extension required in [F11] is supplied by future null completeness, so its local variation data also exist slightly beyond ${\lambda }_f$.

Apply [F7] to the open future set $F$. The equality in step 2.1 shows that $E^{+}(S)$ is closed and is a topological three-manifold without boundary.

Let $p\in E^{+}(S)$ and choose $s\in S$ with $s\le p$. If $s=p$, use the zero-length segment at $s$. If $s\mathrm{\ne }p$, [F4] supplies a maximizing causal geodesic $\gamma$ from $s$ to $p$. Since $p\mathrm{\notin }{I}^{+}(S)$, $s\ll ̸p$, so the maximizing length is zero; the nonconstant affine geodesic has constant squared speed by [F5], and therefore its future causal tangent is null. Rescale its affine parameter by a positive constant so that $g(\dot{\gamma }(0),N_s)=-1$, and translate the parameter so that its initial value is $0$.

Every point $r$ on this segment satisfies $s\le r\le p$. If $r\in I^{+}(S)$, some $s^{\mathrm{\prime }}\in S$ satisfies $s^{\mathrm{\prime }}\ll r\le p$; the push-up rule [F15] then gives $s^{\mathrm{\prime }}\ll p$, contradicting $p\mathrm{\notin }{I}^{+}(S)$. Thus the entire segment lies in $E^{+}(S)$.

The initial null tangent $\mathrm{\ell }=\dot{\gamma }(0)$ is normal to $S$. Suppose otherwise. Then some $X\in T_{s}S$ has $g(\mathrm{\ell },X)\mathrm{\ne }0$; since $S$ is spacelike, $g(X,X)>0$. Choose a smooth curve $\eta (u)\subset S$ with $\eta (0)=s$ and ${\eta }^{\mathrm{\prime }}(0)=X$. The joint exponential map lemma [F13] gives a smooth two-point inverse near $(s,s)$ whose differential in local tangent-bundle coordinates at the diagonal sends basepoint/endpoint variations $(X,\mathrm{\ell })$ to $\mathrm{\ell }-X$. Therefore, after identifying nearby tangent spaces by a smooth local frame, $v(u,\delta ):={\exp }_{\eta (u)}^{-1}(\gamma (\delta ))=\delta \mathrm{\ell }-uX+o(\mathrm{\mid }u\mathrm{\mid }+\delta )$. Choose a small nonzero $a$ with $ag(\mathrm{\ell },X)>0$ and $g(\mathrm{\ell }-aX,\mathrm{\ell }-aX)=-2ag(\mathrm{\ell },X)+a^{2}g(X,X)<0$. With $u=a\delta$, $v(a\delta ,\delta )\mathrm{/}\delta \to \mathrm{\ell }-aX$, so for sufficiently small $\delta >0$ the vector is timelike; its future orientation follows by continuity from the future null vector $\mathrm{\ell }$ for small $\mathrm{\mid }a\mathrm{\mid }$. By [F14], $\eta (a\delta )\ll \gamma (\delta )$. The remaining segment $\gamma {\mathrm{\mid }}_{[\delta ,{\lambda }_p]}$ is causal; [F15] gives $\eta (a\delta )\ll p$, contradicting $p\mathrm{\notin }{I}^{+}(S)$. Hence $\mathrm{\ell }\perp T_{s}S$.

Return to $p\in E^{+}(S)$ and its normal generator from steps 3.2, 4.1, and 5.1, with endpoint parameter ${\lambda }_p$. If a focal instant ${\lambda }_f$ occurred strictly before ${\lambda }_p$, [F11] would give a point $r=\gamma (\lambda )$ with ${\lambda }_f<\lambda <{\lambda }_p$ and $r\in I^{+}(S)$. Since $r\le p$, [F15] would imply $p\in I^{+}(S)$, a contradiction. Hence ${\lambda }_p\le {\lambda }_f\le L$; for $p\in S$ the parameter is $0$ and the same bound holds.

Choose finitely many relatively compact coordinate patches covering $S$, each contained in a patch with the two smooth normalized null normal fields from step 1.2. Their compact closures $K_j$ exist by choosing smaller Euclidean chart balls and applying Heine–Borel [F19] and the continuous-image theorem [F17]. For each $K_j$ and each branch ${\mathrm{\ell }}_{j,\pm }$, the parameter domain $K_j\times [0,L]$ is compact by [F22], and the map $(s,\lambda )\mapsto {\exp }_s(\lambda {\mathrm{\ell }}_{j,\pm }(s))$ is continuous on it by [F12], [F16] and future completeness. Its image is compact by [F17]. By [F18], the finite union $C$ of these compact images is compact. Step 6.1 shows that every $p\in E^{+}(S)$ lies in $C$.

By step 3.1, $E^{+}(S)$ is closed in $M$. Since step 7.1 places it inside the compact set $C$, it is closed in $C$ and hence compact by [F18]. This proves the claim.