Shifting a temporal function permits a controlled compact restriction

Use [F4] to refine the given coordinate cover to finitely many compact chart pieces $C_{\mathrm{\ell }}$ covering $Q$. The ambient smooth manifold is Hausdorff by [F7], so its compact subset $Q$ is closed. Set $K_{\mathrm{\ell }}=C_{\mathrm{\ell }}\cap Q$. By [F7], $K_{\mathrm{\ell }}$ is closed in compact $C_{\mathrm{\ell }}$ and hence compact. Discard empty intersections; the remaining $K_{\mathrm{\ell }}$ still cover nonempty $Q$ and are all subsets of $Q$. Each restricted chart is a homeomorphism onto its coordinate image: an open cover of that image pulls back to an open cover of $K_{\mathrm{\ell }}$, whose finite subcover pushes forward to a finite subcover of the image. Thus the coordinate images are compact, and [F7] identifies their Euclidean metric subspaces as compact metric spaces. Restrictions of continuous real functions remain continuous by [F7]. Applying [F4] on each such image and then taking the minimum and maximum of the finitely many attained extrema gives the extrema on $Q$, attained at points of $Q$. The continuous positive function $h_{\ast }=-g_{\ast }^{-1}(d\tau ,d\tau )$ therefore has $0<m={\min }_Q{h}_{\ast }\le {\max }_Q{h}_{\ast }=M<\mathrm{\infty }$. The fixed derivatives of $\tau$ are bounded on these pieces. By [F5], metric convergence gives uniform inverse-metric convergence, hence $h_j=-g_j^{-1}(d\tau ,d\tau )\to h_{\ast }$ uniformly on $Q$. For late $j$, $m\mathrm{/}2\le h_j\le M+1$, so $a_0=\sqrt{m\mathrm{/}2}\le {\alpha }_j\le a_1=\sqrt{M+1}$. Moreover [F3] gives $d\tau (Z)>0$, and the persistence lemma in [F5], applied to $(g_{\ast },Z,d\tau ,Q)$, identifies these timelike gradients as past directed relative to $Z$. The common orientation hypothesis makes this precisely the orientation used for the future normals $n_j$.

By [F4], the fixed smooth function $\tau$ attains finite extrema on nonempty $Q$. Thus $s=\tau +1-{\min }_Q\tau$ satisfies $1\le s\le T=1+{\max }_Q\tau -{\min }_Q\tau$, with $ds=d\tau$. Put $V_j=-{\alpha }_j^{-1}{\mathrm{grad}}_{g_j}\tau$ and $v_j=-g_j(n_j,V_j)$. By [F3] and step 1.1, $V_j$ is future unit timelike. Uniform inverse-metric bounds, fixed first derivatives of $\tau$ and ${\alpha }_j\ge a_0$ bound its coordinate components. The assumed component bounds for $n_j$ on every actual $\mathrm{\partial }{M}_j$, together with the uniform metric bounds, give a common $B\ge 1$ with $v_j\le B$ there. On the same finite compact cover, metric $C^2$, inverse-metric $C^0$ and fixed shifted-function $C^3$ norms are uniformly bounded.

Apply [F1] under [A1] with temporal function $s$ to $M_j$ itself, using the common time orientation and the smooth actual boundary from the hypotheses. Its entire nonnegative portion is compact and contained in $Q$, since $s\ge 1$ on $M_j\subset Q$. The coefficient and temporal-norm hypotheses follow from steps 1.1 and 2.1. With the common exponent $\mu >4$, the theorem gives ${\sup }_{M_j}{s}^{\mu }{v}_j\le 2\max \{2T^{\mu },T^{\mu }B\}=:C$. Its boundary supremum is bounded by $T^{\mu }B$, or is zero when the boundary is empty. Since $s\ge 1$, $v_j\le C$ throughout $M_j$, and $-g_j(n_j,-{\mathrm{grad}}_{g_j}\tau )={\alpha }_j{v}_j\le a_{1}C$. For a compact restriction the same argument includes every artificial cut in the actual boundary and uses its assumed normal bound. This proves all the assertions.