Small outward normal graphs have controlled inward null initial jets

By [F3] and the supplied extension across $S$, choose $\eta >0$ so that $F_{ϵ}$ is admissible and takes values in the ambient defining-pair tube for $\mathrm{\mid }ϵ\mathrm{\mid }<\eta$. For the latter restriction, the inverse image of that open tube under the smooth graph map is open and contains $\{0\}\times S$; finitely many product neighborhoods of the compact zero section give a common parameter interval. Fix $0<{ϵ}_0<\eta$ and put $P=[0,{ϵ}_0]\times S$, with its product topology. By [F5], $P$ is compact. If $S$ is empty all conclusions are vacuous, so assume $S$ nonempty for extrema below.

At $ϵ=0$, the ambient coordinates of $F_0(y)$ are $(y,0,0)$ and [F1], [F3] give ${\mathrm{\partial }}_{ϵ}(u\circ F_{ϵ}){\mathrm{\mid }}_0=0$ and ${\mathrm{\partial }}_{ϵ}(v\circ F_{ϵ}){\mathrm{\mid }}_0=\varphi (y)$. The second parameter derivatives of these smooth scalar functions are continuous on compact $P$, so [F6] bounds their absolute values uniformly. Apply [F8] with $k=1$ on each parameter segment to get $u\circ F_{ϵ}=O({ϵ}^2)$ and $v\circ F_{ϵ}=ϵ\varphi +O({ϵ}^2)$ uniformly on $S$. Adding and subtracting gives the two position estimates. By [F6], $m={\min }_S\varphi >0$. Thus, after shrinking the positive parameter range, both $a(F_{ϵ}(y))$ and $b(F_{ϵ}(y))$ are at least $mϵ\mathrm{/}2$, so [F1] puts $S_{ϵ}$ in the exterior.

Apply [F4] to the smooth first graph data $(y,ϵ\varphi ,ϵd\varphi )$ and restrict the fixed smooth normal from [F2]. The fields $n_{ϵ},{\nu }_{ϵ},V_{ϵ}$ are jointly smooth, with $V_0=n-\nu$. The gauge and future-null assertions follow from [F4], with $V_{ϵ}=2{\underset{‾}{L}}_{ϵ}$. The scalar functions $d{a}_{F_{ϵ}(y)}(V_{ϵ}(y))$ and $d{b}_{F_{ϵ}(y)}(V_{ϵ}(y))$ are smooth; at zero they equal $-2$ and $0$ by [F1]. Their first parameter derivatives are continuous and bounded on $P$ by [F6]. The $k=0$ bound of [F8] gives the two uniform $O(ϵ)$ derivative estimates.

In the induced bundle charts of [F7], each initial-data map has as components the smooth position coordinates of $F_{ϵ}$ and the smooth vector components supplied in step 2.2. Hence both maps are smooth on an open parameter interval containing $[0,{ϵ}_0]$, and their restrictions to $P$ are continuous. Since $P$ is compact, [F6], claim 1, makes each image compact in $TM$. The maps at $ϵ=0$ have vectors $n-\nu$ and $(n-\nu )\mathrm{/}2$, respectively, so these limiting data are included. This proves the stated compactness for both gauges, together with the estimates of steps 2.1 and 2.2.