Compact estimates for arbitrary sequences of individually existing slices

Discard finitely many terms so that $q_j\in [{\delta }_0,1]$, where ${\delta }_0=\delta \mathrm{/}2>0$, and put $g_j=g_{q_j}$, $g_{\ast }=g_{q_{\ast }}$. Apply the indexed bounds in [F1] only to these existing pairs. Invoke the indexed-pair clause of the controlled-cut supplier in [F4] with the remaining sequence indices and their supplied pairs $(q_j,{\mathrm{\Sigma }}_j)$; equal parameter values need not have equal sections. The base has compact core and spherical inner boundary by [A1] and the product in [F4]. The candidate-independent background conclusion in [F4] gives the actual original reference as $\eta =a(r,y)$ with $\mathrm{\mid }a\mathrm{\mid }\le A$ in its invariant boost tube, without a tangent-alignment assumption. Its positive original stationary lapse and future-timelike $K$ give the reference transversality and orientation required by the controlled-cut input. The candidate Hawking-flow graphs have $\eta =a(r,y)+b_j(r,y)$ with $\mathrm{\mid }{b}_j\mathrm{\mid }\le B$ by [F1]. Their uniformly spacelike smooth normal graphs on a smaller common collar bound their normal components in fixed smooth ambient frames in a fixed physical neighborhood $U$ of $S$. Thus every collar hypothesis of the controlled-cut lemma is met, including agreement of the complete boost model with the physical normal tube. It supplies common inner base cuts with bounded normals.

On the normalized end, [F1] gives uniform slope decay and [F5] gives uniform metric convergence. For $F_j^q=t_{q_j}-{\tilde{u}}_j$, the graph conormal normalization is $n_j=-{\mathrm{grad}}_{g_j}{F}_j^q\mathrm{/}{s}_j$. Since $d{F}_j^q(T)=c_{q_j}$, this yields $N_j=c_{q_j}\mathrm{/}{s}_j$, exactly the formula required in [F5]. With $d_{q_j}\ge 1$, the indexed-pair end-anchor clause gives one common threshold ${\rho }_{\ast }$, bounded end normals, and $N_j\ge 1\mathrm{/}8$ at any fixed original base point $x_{\ast }$ with ${\delta }_0\mathrm{\mid }{x}_{\ast }\mathrm{\mid }\ge {\rho }_{\ast }$. Choose such an $x_{\ast }$ once. The lemma also bounds the original-coordinate normals on every sufficiently distant fixed end cut. This uses uniform normalized metric and slope control, rather than slope control alone.

Given the compact target base set, choose two nested connected compact cut domains ${\mathrm{\Omega }}_1\subset {\mathrm{\Omega }}_0^{\circ }\subset X$ containing it and $x_{\ast }$ in ${\mathrm{\Omega }}_1^{\circ }$, using smaller inner cuts and larger outer cuts for ${\mathrm{\Omega }}_0$. The controlled-cut lemma in [F4] supplies their smoothness, compactness and exact two-component boundaries. Steps 1.1 and 2.1 bound all actual boundary normals. Compact height bounds [F1] and the restriction lemma [F3] enclose the larger restrictions $M_j=\{(u_j(x),x):x\in {\mathrm{\Omega }}_0\}$ in one fixed compact stationary tube $Q_{+}$. Choose this tube with a time margin; choose a smaller tube $Q\subset Q_{+}^{\circ }$ containing all graphs over ${\mathrm{\Omega }}_1$. Enclose $Q_{+}$ in a relatively compact open stationary tube $B\subset \mathbb{R}\times X$. Only the supplied graphs are restricted; no cut is treated as absent from the actual boundary.

Use the smooth transformed time orientations from the energy supplier [F6]. At $q_{\ast }$ its orientation field is a fixed smooth future timelike field $Y$ for $g_{\ast }$; by smoothness and compactness it is also future timelike for all late $g_j$ near $\bar{B}$, with the same future convention. Fix any compact $C\subset B$ and any eligible future $g_{\ast }$-timelike Lipschitz curve $\gamma :[a,b]\to C$ with $g_{\ast }(\dot{\gamma },\dot{\gamma })\le -{ϵ}_{\gamma }<0$ almost everywhere. Its coordinate speed is essentially bounded in a finite chart cover. Uniform metric convergence therefore makes the same curve $g_j$-timelike for all sufficiently large $j$ (depending on the curve); uniform compact cone margins and the common orientation make it future directed. For $a_j=-g_j(\dot{\gamma },\dot{\gamma })$ and $a_{\ast }=-g_{\ast }(\dot{\gamma },\dot{\gamma })$, the elementary estimate $\mathrm{\mid }\sqrt{a_j}-\sqrt{a_{\ast }}\mathrm{\mid }\le \sqrt{\mathrm{\mid }{a}_j-a_{\ast }\mathrm{\mid }}$ bounds the integrand difference by $C_{\gamma }\mathrm{\parallel }{g}_j-g_{\ast }{\mathrm{\parallel }}_{C^0(C)}^{1\mathrm{/}2}$ almost everywhere. Integrating over the finite interval proves convergence of these fixed-curve lengths.

Extend each candidate normal and lapse by stationary translation, as in [F3], and put $G_j=t-u_j$. The identity $d{G}_j=-g_j(n_j,\cdot )\mathrm{/}{N}_j$ gives the needed pointwise length inequality: decompose a future timelike velocity as $v=a{n}_j+w$ with $w\perp n_j$ and $a>0$. Positivity on $n_j^{\perp }$ gives $a^2=-g_j(v,v)+g_j(w,w)\ge -g_j(v,v)$. Consequently $\sqrt{-g_j(\dot{\gamma },\dot{\gamma })}\le N_{j}d{G}_j(\dot{\gamma })\le C_{{\delta }_0}d{G}_j(\dot{\gamma })$, using the global upper lapse bound in [F1]. Integration bounds the length by $C_{{\delta }_0}{\mathrm{osc}}_C{G}_j$. Compact stationary height bounds on the projection of $C$, and bounded $t$ on $C$, bound this oscillation uniformly in $j$ and in the curve. Pass to the limit using step 4.1. Thus every compact $C\subset B$ has the required uniform $g_{\ast }$-length bound, although its admissible late index may depend on the curve. The exact all-compact hypothesis of [F2] holds. Apply it to $Q_{+}$ to obtain a fixed smooth temporal function $\tau$ on a neighborhood of $Q_{+}$.

Shift $\tau$ to $s=\tau +1-{\min }_{Q_{+}}\tau \ge 1$ on $Q_{+}$. The metrics converge smoothly there, the time orientations agree by step 4.1, and every $M_j$ is compact, smooth, spacelike maximal and contained in $Q_{+}$ with all actual boundary normals bounded by step 3.1. The weighted restriction lemma [F3] therefore applies to the entire nonnegative-$s$ portion, which is exactly $M_j$, and bounds its temporal normal angle. Its temporal direction is uniformly timelike on the compact tube. A future unit normal of bounded angle to such a direction has bounded coordinate components: in an adapted orthonormal frame write $n=vV+w$ with $g_j(w,V)=0$ and $g_j(w,w)=v^2-1$. The bounded angle $v$ and the compact background frame bounds control all components. Hence the candidate normals have one common coordinate bound throughout the larger restrictions, including a buffer around $Q$.

The stationary level lemma [F3] makes each global graph closed, with positive lapse from [F1]. The compact background family, normal bound on $Q_{+}$ and the buffer $Q\subset Q_{+}^{\circ }$ verify the graph-geometry lemma, which supplies uniform local temporal graph bounds of every fixed finite order, induced ellipticity and second-form bounds. Apply its bounded connected-cover assertion to $L_j=\{(u_j(x),x):x\in {\mathrm{\Omega }}_1\}$, a connected compact portion containing both the target and the anchor. All its chart neighborhoods lie away from the actual boundaries of $M_j$. This produces a uniformly bounded connected chain with the larger buffered chart controls; no uniform lower stationary lapse has been used to obtain these temporal charts.

For each candidate, $T$ is Killing and its mean curvature is zero. Apply the geometric variation identity [F6] to obtain ${\mathrm{\Delta }}_{h_j}{N}_j=V_j{N}_j$ with $V_j=\mathrm{\mid }{k}_j{\mathrm{\mid }}_{h_j}^2+\mathrm{Ric}(g_j)(n_j,n_j)$. The electrovacuum energy supplier [F6] applies to the same regular Harrison family and supplied future unit normal, so $V_j\ge 0$. The second-form bounds from step 7.1 and compact smooth ambient Ricci bounds paired with the bounded normals give a common upper bound on $V_j$ on the buffered chain. The lapses are positive by [F1] and have the actual anchor $N_j(x_{\ast })\ge 1\mathrm{/}8$ from step 2.1. Every hypothesis of the anchored lapse and stationary-transfer lemma [F3] is now satisfied. It yields a common positive lower lapse bound on $L_j$ and transfers the temporal derivative bounds to stationary heights of every fixed finite order on the target base set. Together with step 6.1 this proves all three asserted estimates for the arbitrary existing sequence.