Openness via the weighted implicit-function branch and smooth boundary bootstrap

Put $\beta =-3\mathrm{/}4$, $\bar{h}=q_{\ast }^{-2}{h}_{\ast }$ and $\bar{V}=q_{\ast }^2{V}_{\ast }$. By [F3], the reference is a connected smooth exterior, $\bar{h}-\delta \in C_{-1}^{3,\alpha }$, $0\le \bar{V}\in C_{-4}^{2,\alpha }$, and X and $Z=C_{\beta -2}^{2,\alpha }$ are Banach. Thus [F4] makes $\bar{L}={\mathrm{\Delta }}_{\bar{h}}-\bar{V}:X\to Z$ a bounded isomorphism. Also [F3] makes multiplication by $a_{\ast }=-g_{q_{\ast }}(Y,n_{\ast })$ a bounded isomorphism of X. The derivative is $A_{\ast }=q_{\ast }^{-2}\bar{L}{M}_{a_{\ast }}$. Its inverse on a target value f is $M_{a_{\ast }^{-1}}{\bar{L}}^{-1}(q_{\ast }^{2}f)$, a bounded composition. This verifies actual invertibility, not merely zero Fredholm index.

The admissible map from [F3] is C1 on an open neighborhood in $\mathbb{R}\times X$, with value zero at $(q_{\ast },0)$ because the reference is maximal. Apply [F6] with parameter space the real line, unknown space X and target Z, using the bounded inverse from step 1.1. It gives a unique local C1 branch $w(q)$ with $w(q_{\ast })=0$ and $\mathcal{H}(q,w(q))=0$. Shrink the neighborhood so its graphs stay embedded, spacelike and fixed pointwise on S as in [F14]. If an endpoint metric extension was used, retain only the physical parameters.

Each w(q) initially lies in C4-alpha up to S. On every compact core or finite end chart, [F7] gives a smooth uniformly elliptic quasilinear equation with zero boundary height on S. The chain [F8]–[F11] applies: the interior smoothness is first established by classical difference quotients; tangential quotients at the boundary obey the exact Dirichlet Schauder theorem; the elliptic equation then recovers the other normal derivatives. Hence each w(q) is smooth up to S and throughout the interior. No prior smoothness is assumed to apply the classical quotient estimates. This changes no weighted-norm or boundary condition of the branch.

On the end the transverse field is T, so in the fixed original spatial coordinates the new original-time height is $u_q(x)=u_{\ast }(x)+w(q)(x)$. Put $c_q=d_q\mathrm{/}(4q)$. By [F3], $c_{q_{\ast }}>0$; [F16] makes $d_q$ a continuous polynomial in q. Since $q_{\ast }>0$, shrink the parameter neighborhood so q, $d_q$, $c_q$ and their reciprocals have common positive bounds. By [F15], $u_{\ast }(x)=c_{q_{\ast }}^{-1}{\tilde{u}}_{\ast }(q_{\ast }x)$. The assumed reference estimates therefore give $\mathrm{\mid }{D}_x^j{u}_{\ast }\mathrm{\mid }\le C{r}^{-3\mathrm{/}4-j}$ for $0\le j\le 4$, by constant coordinate scaling; no fourth-derivative Hölder seminorm is needed for this inference. The weighted norm of w(q) in [F13] gives $\mathrm{\mid }{D}_x^{j}w(q)\mathrm{\mid }\le C{r}^{-3\mathrm{/}4-j}$ for the same orders, locally uniformly in q by the weighted-norm continuity of the branch. Adding these estimates and using ${\tilde{u}}_q(x_q)=c_q{u}_q(x_q\mathrm{/}q)$, $r_q=qr$ and $D_{x_q}^j{\tilde{u}}_q=c_q{q}^{-j}(D_x^j{u}_q)(x_q\mathrm{/}q)$ gives $\mathrm{\mid }{D}_{x_q}^j{\tilde{u}}_q\mathrm{\mid }\le C{r}_q^{-3\mathrm{/}4-j}$ for $0\le j\le 4$, with locally uniform constants. In particular the normalized end height tends to zero, giving the required end conditions before the global projection is invoked.

Since w(q) tends to zero in X, shrink the parameter neighborhood again to satisfy the fixed-reference smallness conditions in [F5]. The candidates are now smooth by step 3.1, maximal by step 2.1 and have the required end data by step 3.2. The supplied lapse, boundary return, properness and one-sheet projection result therefore turns each into a global stationary graph without presupposing that conclusion or transformed global hyperbolicity. Its boundary remains S. This proves the asserted nearby smooth-slice construction and relative openness under the supplied reference hypotheses.