Uniform compact graph and positive-lapse estimates

If $\delta >1$, the parameter interval is empty and there are no candidate points or parameter sequences to estimate. If $\delta =1$, the supplied family consists of the one smooth graph $u_1$ for the stationary metric $g_1$, with smooth future normal and positive lapse $N_1$ as assumed in [A1]. The level-function helper in [F3] applies to this graph: ${\tau }_1=t-u_1(x)$ is smooth with past timelike gradient on the whole exterior, the graph is closed there, and ${\mathrm{grad}}_{g_1}{\tau }_1=-n_1\mathrm{/}{N}_1$ on the graph. For nonempty compact $K$, [F8] makes its graph image compact and gives finite bounds on its smooth coordinate-normal components and a strictly positive minimum $c_K$ of $N_1$. Thus every sequence with parameters in $[1,1]$ has $q_j=q_{\ast }=1$, and taking ${\tau }_{\ast }={\tau }_1$ gives $-g_1(n_1,-{\mathrm{grad}}_{g_1}{\tau }_{\ast })=1\mathrm{/}{N}_1\le c_K^{-1}$. This function is available on a neighborhood of every compact stationary tube. Empty $K$ requires no estimate. Still for $\delta =1$, place the compact graph image over $K$ in the interior of a larger compact stationary tube, with a further buffered tube inside the exterior. Extend its normal by stationary translations as in the level-function helper. With $h_1$ the induced metric of the graph pulled back to the base, $e=d{t}^2+h_1$ is a smooth auxiliary positive-definite metric; [F8] bounds the extended normal in this metric on the buffered tube. Apply the interior geometry helper in [F3] to the constant metric family consisting of $g_1$ and the single closed smooth spacelike maximal graph $u_1$, with these buffered tubes and this normal bound. It gives the claimed uniform local temporal-graph charts and finite-order estimates for the singleton. For the stationary estimates, cover $K$ by finitely many smaller coordinate balls with larger compactly contained chart closures. For every fixed finite $m\ge 0$, all derivatives of the smooth $u_1$ through order $m+1$ are bounded on each larger closure by [F8]. Applied to each order-$m$ derivative on the convex open ball, the mean-value inequality in [F8] bounds its difference at two points by $L_m\mathrm{\mid }x-y\mathrm{\mid }$, hence its $\alpha$-Hölder quotient by $L_m\mathrm{diam}(B{)}^{1-\alpha }$ for $0<\alpha <1$. Taking all lower-order suprema and the finitely many chart bounds gives the full $C^{m,\alpha }(K)$ estimate, including $m=4$, in the compact-set convention of the anchored stationary-transfer helper in [F3]. The constants may depend on this supplied singleton; no family of candidates at parameters below one has been introduced. This proves all assertions for $\delta =1$.

For the rest of the argument assume $0<\delta <1$, so the interval hypotheses of [F1] and [F2] match their cited suppliers. Choose two nested smooth compact reference annuli containing K, with inner cuts in the complete Hawking collar and outer cuts far in the common normalized end. By [A1], $H_g{\mathrm{\mid }}_C=0$ on the actual physical section and ${\mathrm{\Sigma }}_1=C\cup S$, so [F4] and the common collar in [F1] apply to this same reference section. Apply [F4] to the fixed original metric and section: the section is smooth and spacelike through $S$, its exterior is the positive outward wedge, its stationary lapse is positive, and $K$ is future timelike there by [F1]. Thus the invariant boost coordinates give $K={\mathrm{\partial }}_{\eta }$ and $T={\mathrm{\partial }}_{\eta }-{\mathrm{\Omega }}_H{Z}_S$. The candidate-independent background branch in [F4] gives the reference graph $\eta =a(r,y)$ with $\mathrm{\mid }a\mathrm{\mid }\le A$, allowing its actual spacelike tilt. Here $a$ is boost time. Its conormal satisfies $d(\eta -a)(K)=1$ and is past timelike, so positive original stationary lapse gives $d(\eta -a)(T)=1+{\mathrm{\Omega }}_H{Z}_{S}a>0$, the reference transversality required by the cut helper. This is one fixed reference graph, so its bound is independent of $q$. The candidate Hawking-height bound in [F1] then gives $\eta =a+b_q$ with $\mathrm{\mid }{b}_q\mathrm{\mid }\le B$. The cut helper in [F3] therefore makes their inner candidate boundaries lie in the common physical region where horizon normals are uniformly controlled. At the pole, $f(p)=-g(T,T)(p)=0$, so the denominator definition gives $d_q=(1+q{)}^2+(1-q{)}^2\psi (p{)}^2\ge 1$. Together with the uniform normalized metric convergence in [F6] and the differentiated graph-slope decay in [F1], this verifies all end hypotheses of the cut helper. Let ${\rho }_{\ast }$ be its common normalized threshold, choose a fixed original base point $x_{\ast }$ with $\delta \mathrm{\mid }{x}_{\ast }\mathrm{\mid }\ge {\rho }_{\ast }$, and choose both outer cuts beyond $x_{\ast }$. The same helper bounds the outer-boundary normals. The level-set helper in [F3] makes the graph restrictions smooth compact hypersurfaces with these actual boundaries. The compact-height supplier applies with the two-sided access premise in [A1], in the original metric and exterior and using the same physical end coordinate as the candidate normalization. Its compact height bounds over the fixed larger reference annulus enclose every restriction in one fixed stationary tube, separated from S; choose still larger buffered tubes for [F2].

Fix a sequence $q_j\to q_{\ast }\in [\delta ,1]$. Apply [F2] to a tube containing the entire larger graph restriction, not merely the desired target points. Shift its supplied temporal function by a fixed constant so that it is at least one on that whole tube. The normal-angle helper in [F3] applies to the compact restrictions themselves: their entire nonnegative-height portions are exactly those restrictions, their actual boundaries are the controlled cuts in step 1.2, and their geometry and temporal coefficients have common bounds on the fixed tube. It gives a uniform angle and hence coordinate-normal bound throughout the larger restriction. No artificial cut or boundary term has been discarded.

Positivity of stationary lapse in [F1] makes each global stationary level function temporal by the first helper in [F3]. Therefore every local future timelike fibre meets a candidate at most once. Apply the interior geometry helper on a buffered region between the nested restrictions: it provides uniform local temporal graph radii, derivative bounds and a uniformly bounded finite connected chart cover joining the target to a fixed distant end anchor. Its volume/packing argument supplies the bounded number of charts; compactness alone is not used to assume a uniform chart count. The end cut helper in [F3], using the uniform normalized metric convergence in [F6], the graph-slope decay in [F1] and $d_q\ge 1$ established in step 1.2, supplies $N_q\ge 1\mathrm{/}8$ beyond a common normalized radius ${\rho }_{\ast }$. The fixed original base point $x_{\ast }$ chosen inside both restrictions in step 1.2 satisfies $\mathrm{\mid }q{x}_{\ast }\mathrm{\mid }\ge {\rho }_{\ast }$ for every $q\in [\delta ,1]$, so it is one common anchor for the chart chain.

The lapse equation in [F1] has potential $V_q=\mathrm{\mid }{k}_q{\mathrm{\mid }}_{h_q}^2+\mathrm{Ric}(g_q)(n_q,n_q)$. The energy-condition result [F5] applies under the regular mass-one vacuum, original-metric maximality of the actual physical exterior section, and selected-Harrison-family hypotheses in [A1]; its timelike Ricci inequality and the positive-definite induced metric give $V_q\ge 0$ for these same future unit normals. The normal and graph-regularity bounds from step 3.1 bound this potential and all metric coefficients required by Harnack on the finite chain. The anchored propagation helper in [F3] yields $N_j\ge c>0$ on the target and on the chosen chain. Apply the full coordinate-transfer conclusion of that same helper: take $L_j$ to be the connected compact graph portion over the inner reference annulus containing both $K$ and the fixed end anchor from step 3.1. Its entire lift of $K$ is contained in $L_j$, and step 2.1 supplies the buffered normal bound, while step 3.1 supplies the bounded connected cover with all its chart buffers. The positive lapse equation and nonnegative bounded potential just verified match the remaining hypotheses. Thus stationary projection has smooth local inverses on base balls of a common radius with uniformly bounded inverse differentials. The helper obtains every higher inverse derivative by differentiating the inverse identity, then composes with the controlled temporal graphs. It therefore yields the full stationary estimates $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{C^{m,\alpha }(K)}\le C_{K,m,\alpha }$ for every finite $m\ge 0$ and $0<\alpha <1$, in its fixed-chart compact-set norm; in particular it yields the stated fourth-order estimate. No stationary ellipticity or uniform stationary chart radius was assumed before this positive lower lapse bound.

The argument applies to an arbitrary sequence of admissible candidates whose parameters converge to any $q_{\ast }\in [\delta ,1]$. If any asserted uniform compact bound failed for the full family, choose a violating sequence. Its parameters lie in the bounded interval $[\delta ,1]$, so [F7] gives a convergent parameter subsequence. Its limit remains in that interval: a limit below $\delta$ would, for $ϵ=(\delta -q_{\ast })\mathrm{/}2$, force late terms below $\delta$, and a limit above $1$ would similarly force late terms above $1$, contradicting their bounds. Steps 1.2, 2.1, 3.1 and 4.1 give a uniform bound on its tail, contradicting the chosen divergence. Together with the singleton and empty-interval cases in step 1.1, this proves the parameter-uniform claims for every $\delta >0$, while the raw temporal-gradient pairing assertion is exactly the shifted weighted conclusion in step 2.1 before the coordinate transfer.