Complete two-ended geometry and decay of the reflected data

The compact-complement hypothesis is an explicit premise for the abstract positive half. For the separate canonical-section application, assume the full supplier hypotheses specified in F4 and use its conditional closure, compact-change and unchanged-end conclusions. Let $L$ be a compact ambient set containing the region of change, and take $R$ sufficiently large that the common stationary tail $r>R$ misses $L$. Write $D_R$ for the original compact radial truncation. The new truncation $B_R=(C_0\cup S)\setminus \{r>R\}$ is contained in $L\cup D_R$: every point outside $L$ is unchanged, and outside the removed tail is therefore in $D_R$. The set $C_0\cup S$ is closed in the ambient manifold because it is the supplied ambient closure; the common tail is relatively open in it. Thus $B_R$ is ambient closed. To check compactness directly, represent any relative open cover of $B_R$ by ambient open sets and add the ambient open complement of $B_R$. This covers each of the compact sets $L$ and $D_R$; combining their finite subcovers and discarding the complement gives a finite subcover of $B_R$. The boundary is $S$ together with the unchanged end sphere $r=R$, since the closure is a smooth manifold with boundary $S$ and the additional cut is made in the smooth end chart. These truncations exhaust the closure. This proves the canonical compact-complement application without requiring an isotopy or any spacetime development of the double.

In the copying charts the metric coefficients on the two ends coincide, and the $K$ coefficients have opposite signs. Every Cartesian derivative is therefore the same or its negative on the two copies. All the order-six metric and order-five second-fundamental-form estimates carry over unchanged. For $r\ge 1$, dropping derivatives gives $h-\delta =O_4(r^{-1})$, and $r^{-3-\mathrm{\mid }\beta \mathrm{\mid }}\le r^{-2-\mathrm{\mid }\beta \mathrm{\mid }}$ for $\mathrm{\mid }\beta \mathrm{\mid }\le 3$ gives $K=O_3(r^{-2})$. For the separate canonical application, assume all supplier hypotheses in F3 and F4. F4 supplies the original metric bound and its preservation on the unchanged end of the splice; F3 supplies the original extrinsic-curvature bound, preserved on that same unchanged end. The end-decay deduction requires no maximality assumption.

For each sufficiently large $R$, let $H_R$ be the image in $\widehat{C}$ of the two copies of $B_R$. It is compact: an open cover pulls back to an open cover of each compact copy, and the two finite subcovers together cover $H_R$. Its complement is precisely the disjoint union of the two copied tails $r>R$. The $H_R$ exhaust the double. The two connected halves meet along the nonempty seam $S$, so their union is connected: a separation would restrict to a separation of each half, and their common points force them into the same member. F1 and the inherited choice convention give the smooth boundaryless Riemannian double. Each tail is a Euclidean exterior and its larger radial tails stay connected, so this compact exhaustion has exactly two Euclidean ends and no additional end.

By the zeroth-order metric decay, choose $R$ large enough that $h(v,v)\ge \frac{1}{2}\mathrm{\mid }v{\mathrm{\mid }}^2$ on both tails $r\ge R$. Fix $q\in H_R$. Any piecewise $C^1$ curve from $q$ to a point $p$ in one tail with $r(p)>R$ must cross that tail's sphere $r=R$. Its last crossing exists because the parameter interval and that sphere are compact and the sphere is closed. The remaining curve lies in $r\ge R$ in the same tail. On each smooth piece there, $\mathrm{\mid }(r\circ \gamma {)}^{\mathrm{\prime }}\mathrm{\mid }\le \mathrm{\mid }\dot{x}\mathrm{\mid }\le \sqrt{2}\sqrt{h(\dot{\gamma },\dot{\gamma })}$. Integration gives $L_h(\gamma )\ge (r(p)-R)\mathrm{/}\sqrt{2}$. Taking the infimum over all curves, including ones that visit the compact region or the other end, gives $d_h(q,p)\ge (r(p)-R)\mathrm{/}\sqrt{2}$.

Let $(p_j)$ be any $d_h$-Cauchy sequence. Choose $j_0$ such that $d_h(p_j,p_{j_0})<1$ for all $j\ge j_0$, and enlarge the radius in step 3.1 so that $q=p_{j_0}\in H_R$. That step implies that every tail point $p_j$ with $j\ge j_0$ has $r(p_j)<R+\sqrt{2}$; the other such points already lie in $H_R$. Increase a radius $A>R+\sqrt{2}$ to include the finitely many earlier points as well. The whole sequence is contained in the compact set $H_A$. This bound rules out unbounded excursions through either end.

By F5, the topology of the restricted metric $d_h{\mathrm{\mid }}_{H_A\times H_A}$ is the subspace manifold topology on $H_A$. Hence its compactness from step 2.1 is compactness for this restricted metric. By F6 that compact metric space is complete, so the sequence in step 4.1 converges in this same distance to a point of $H_A\subset \widehat{C}$. This is the restriction of the distance on the whole double, and need not equal the intrinsic distance of $H_A$. Every $d_h$-Cauchy sequence thus converges in $\widehat{C}$, which proves the claimed metric completeness.

For the additional lapse assertion, retain the stationary KID hypotheses stated explicitly in the Statement. On the negative copied end F2 gives $N_{-}=-N_{+}\circ I$, so $N_{+}\to 1$ implies $N_{-}\to -1$ in the copying chart. This does not assert that the same stationary Killing field is future timelike at both ends. The metric conclusions proved in the preceding steps do not require these additional Killing-field hypotheses. Together these arguments prove all stated initial-data conclusions, without identifying a domain of outer communications in any development.