Positive-lapse stationary graphs give temporal level functions and closed restrictions

On $\mathrm{\Gamma }$, $dF$ annihilates the graph tangent space and $dF(T)=1$. The covector $-g(n,\cdot )\mathrm{/}N$ also annihilates that space and evaluates to one on $T$, since $N=-g(T,n)$. The tangent space has codimension one and $T$ is transverse to it, so agreement on that space and on $T$ proves $dF=-g(n,\cdot )\mathrm{/}N$. Its metric dual is $-n\mathrm{/}N$, which is past timelike because $n$ is future unit timelike and $N>0$.

The function $F$ is smooth and continuous, so [F4] applied to the closed subset $\{0\}\subset \mathbb{R}$ makes $\mathrm{\Gamma }=F^{-1}(0)$ closed relative to $M$. The graph map $G:X\to M$, $G(x)=(u(x),x)$, is smooth and injective. Its differential sends $v$ to $(du(v),v)$ and is injective. Product projection restricted to its image is a continuous inverse; hence $G$ is a smooth embedding by [F3], applied to manifolds with empty boundary.

Every point $(t,x)$ belongs to the unique translate ${\mathrm{\Phi }}_c(\mathrm{\Gamma })$ with $c=t-u(x)$. Define $n(t,x)=({\mathrm{\Phi }}_c{)}_{\ast }n(u(x),x)$ and $N(t,x)=N(u(x),x)$. These extensions are smooth. Translations preserve the metric, time orientation and $T$, and $F\circ {\mathrm{\Phi }}_c=F+c$. Transporting the covector identity from step 1.1 therefore gives $dF=-g(n,\cdot )\mathrm{/}N$ everywhere on $M$, with past timelike dual $-n\mathrm{/}N$.

The restriction $G{\mathrm{\mid }}_{\mathrm{\Omega }}$ is smooth up to the boundary because it is the composite of the supplied smooth inclusion $\mathrm{\Omega }\hookrightarrow X$ with $G$. Its differential is injective on the full tangent spaces, and projection to $X$ is a continuous inverse onto $\mathrm{\Omega }$ with its subspace topology. Thus [F3] makes $G{\mathrm{\mid }}_{\mathrm{\Omega }}$ an embedding. Equip its image with the smooth structure transported from $\mathrm{\Omega }$. Each boundary chart $\phi$ of $\mathrm{\Omega }$ gives the image chart $\phi \circ (G{\mathrm{\mid }}_{\mathrm{\Omega }}{)}^{-1}$ with exactly the same half-space coordinates. The last coordinate vanishes precisely on $G(\mathrm{\partial }\mathrm{\Omega })$, so the manifold boundary of ${\mathrm{\Gamma }}_{\mathrm{\Omega }}$ is exactly that image. The full tangent spaces of this three-dimensional image lie in the spacelike tangent spaces of $\mathrm{\Gamma }$, so the induced metric remains positive definite, including at the boundary. These are actual restriction boundaries; they have not been discarded.

The product $M$ is a smooth four-manifold, and step 2.1 makes $F$ a smooth temporal function for its given time orientation. By [A1], both choice hypotheses of [F2] hold. Part (b) of that supplier therefore makes $F\circ \gamma$ locally absolutely continuous and strictly increasing on every future timelike curve $\gamma$. This invocation includes the almost-everywhere chain rule and integration step, rather than inferring curve monotonicity from pointwise positivity alone. Two parameter values on $\mathrm{\Gamma }=F^{-1}(0)$ would give equal values of $F$, contradicting strict increase.

Apply the continuous-image assertion in [F4] to the compact space $\mathrm{\Omega }$ and the continuous map $G{\mathrm{\mid }}_{\mathrm{\Omega }}$: the image ${\mathrm{\Gamma }}_{\mathrm{\Omega }}$ is compact. If a family has $\mathrm{\mid }u\mathrm{\mid }\le A$ on the fixed $\mathrm{\Omega }$ with $A\ge 0$, every image lies in $[-A,A]\times \mathrm{\Omega }$. The interval supplier applies with endpoints $-A\le A$, and the finite-product supplier applies to that compact interval and the compact space $\mathrm{\Omega }$. Their product topology is the subspace topology of this tube in the stationary product, so the tube is compact. The same conclusion includes empty $\mathrm{\Omega }$ and $A=0$.