Mean-curvature linearization, weighted regularity and fixed-boundary graph map

The standing model, original maximal section, mass-one normalization and $T{\mathrm{\mid }}_S\equiv ̸0$ are exactly the hypotheses of [F6]. Use its single smooth stationary $W_0$, the same choice specified in [F4], with its normalized contraction and original-chart spatial bound; no equality with a radial integral is used. The original stationary order-six bounds in [F2] and Killing equation in [F8] give $F_{ab}=\frac{1}{2}({\mathrm{\partial }}_a{g}_{b0}-{\mathrm{\partial }}_b{g}_{a0})=O_5(r^{-2})$. The Hodge coefficients, expressed using the inverse metric and volume density, are bounded through six spatial derivatives, with positive-order derivatives of the asymptotically constant coefficients gaining the corresponding powers of $r^{-1}$. Hence $\mathcal{F}=F+i{\star }_{g}F=O_5(r^{-2})$. By [F8], $d\sigma =2{\iota }_T\mathcal{F}$ and $\sigma \to 1$. Integrating its radial derivative from $r$ to infinity gives $\sigma -1=O(r^{-1})$; its positive-order derivatives through order six are those of $d\sigma$, so $\sigma -1=O_6(r^{-1})$. Thus, writing $B_q=2q+(1-q)(1-\sigma )$, on every fixed compact positive-parameter interval $J$ one has $D_q=\mathrm{\mid }{B}_q{\mathrm{\mid }}^2=4q^2+O_5(r^{-1})$, with common constants also after either of the first two parameter derivatives. On a sufficiently distant common end $D_q$ is bounded away from zero. The family and contraction in [F9] give $(W_q{)}_i=(1-q{)}^2(W_0{)}_i-4q(1-q)g_{0i}=O_5(r^{-2})$ and $(W_q{)}_0=d_q-D_q$. Consequently the components of $A_q$ and the scalar ${\bar{B}}_q\mathrm{/}{B}_q$ are bounded through three spatial derivatives, with each positive derivative of order $j$ bounded by $C{r}^{-j}$: differentiate these finite products and reciprocals using the lower bound for $D_q$ and the polynomial formula for $d_q$ in [F10]. The pullback formula in [F7] now yields $F_{q_{\ast }}=O_3(r^{-2})$. The constant coordinate changes in [F3] preserve these orders, giving $F_{q_{\ast }}=O_3(r_{q_{\ast }}^{-2})$, and [F12] gives $F_{q_{\ast }}\in C_{-2}^{2,\alpha }$. No parameter derivative of the electromagnetic square-root factor is used.

The additional positive-parameter conclusion in [F5] applies to the same smooth global primitive from [F6]: the regular model, original maximality, mass-one normalization and $T{\mathrm{\mid }}_S\equiv ̸0$ are the standing hypotheses. Apply [F13] to the existing maximal end graph using the background bounds in [F5]. It gives the missing weighted Hölder control and one further graph derivative. [F14] yields the induced metric and normal estimates; [F15], using step 1.1 and the electrovacuum equations in [F4], gives the potential in $C_{-4}^{2,\alpha }$. Smoothness on the supplied compact part supplies its finite local norms. By [F11], the potential is nonnegative. Constant coordinate scaling gives the displayed $q_{\ast }^{-2}{h}_{\ast }$ estimate in the fixed original chart.

Verify the general stationary-family hypotheses of [F18] for this actual smooth primitive. Put $v_i=(W_q{)}_i$. Expanding [F24] and substituting $(W_q{)}_0=d_q-D_q$ from [F9] gives in the fixed original chart $(g_q{)}_{00}=-\frac{f{d}_q^2}{4D_q},(g_q{)}_{0i}=\frac{d_q}{4}(g_{0i}-\frac{f{v}_i}{D_q}),(g_q{)}_{ij}=\frac{1}{4}(D_q{g}_{ij}+g_{0i}{v}_j+g_{0j}{v}_i-\frac{f{v}_i{v}_j}{D_q})\mathrm{.}$ Step 1.1 gives $D_q=4q^2+O_5(r^{-1})$ and $v_i=O_5(r^{-2})$, also after zero, one or two parameter derivatives on a compact positive interval $J$. Since $D_q$ has a common positive lower bound on the distant end, the identity $D_q^{-1}-(4q^2{)}^{-1}=-(D_q-4q^2)\mathrm{/}(4q^2{D}_q)$ and repeated product differentiation give $D_q^{-1}=(4q^2{)}^{-1}+O_5(r^{-1})$ with the same mixed bounds. In [F2], $f=1+O_6(r^{-1})$, $g_{ij}={\delta }_{ij}+O_6(r^{-1})$ and $g_{0i}=O_6(r^{-2})$; these fields have no parameter dependence. The displayed component formulas therefore give, for $j=0,1,2$, ${\mathrm{\partial }}_q^j((g_q{)}_{00}+c(q{)}^2)=O_5(r^{-1}),{\mathrm{\partial }}_q^j((g_q{)}_{ij}-b(q{)}^2{\delta }_{ij})=O_5(r^{-1}),{\mathrm{\partial }}_q^j(g_q{)}_{0i}=O_5(r^{-2}),b(q)=q,c(q)=\frac{d_q}{4q}\mathrm{.}$ Every spatial-block product involving $v_i$ has order at most $-4$, and every error in the other leading terms contains a factor of order at most $-1$; the finite product and reciprocal rules retain these orders under the stated mixed derivatives. Positivity and smoothness of $b,c$ follow from [F10]. The common second-parameter bounds meet the sufficient alternative in [F18] for its annular $C^{4,\alpha }$ parameter-norm conditions, using [F12] for the spatial Hölder upgrade. Thus rational dependence alone has not been substituted for a decay or parameter-norm estimate.

Choose the field and graph neighborhood from [F17]. It equals the normal near S, so its lapse there is one, and equals T far out. At the supplied pole, $T_p=0$ gives $f(p)=0$, so [F10] yields $d_{q_{\ast }}=(1+q_{\ast }{)}^2+(1-q_{\ast }{)}^2\psi (p{)}^2>0$. The time scaling in [F3] gives $T={\mathrm{\partial }}_t=(d_{q_{\ast }}\mathrm{/}(4q_{\ast })){\mathrm{\partial }}_{t_{q_{\ast }}}$. The normalized metric and normal asymptotics in [F5] and [F14] therefore give $a_{\ast }\to d_{q_{\ast }}\mathrm{/}(4q_{\ast })>0$. [F16] supplies the positive lower bound, weighted inverse multiplier and preservation of zero trace. In particular no vanishing stationary lapse at S is incorrectly inverted.

The common stationary product and fixed end coordinates are [F3]. The jointly smooth metric on the exterior and a common neighborhood of $S$ is [F4]; the reference is smooth, spacelike and maximal by hypothesis. On the tail the mixed bounds in step 2.2 and the decaying reference slope give a common positive lower bound for the induced spatial metric on a smaller parameter interval. On the remaining compact part, continuity preserves spacelikeness after shrinking that interval. These match the general analytic hypotheses of [F18], whose reference height is the constant rescaling of the supplied normalized height in [F3]. Apply it to get $u_{\ast }\in C_{-3\mathrm{/}4}^{5,\alpha }$, ${\mathrm{\Delta }}_0{u}_{\ast }\in C_{-3}^{2,\alpha }$ and the $C^1$ reference residual in weight $-3$. Its separate Harrison application is formulated under the complete Radu GR/SBS contract; that stronger application is not inferred from the present weaker end data. Here the general interface is applied with the actual mixed bounds proved in step 2.2. At $q_{\ast }=1$, $D_1=d_1=4$. The polynomial definitions of $W_q,D_q,d_q$ extend to real $q$ near one. The end bounds of steps 1.1 and 2.2 persist there, and positivity of $D_q,d_q$ persists on the end and on a common neighborhood of the compact remainder by continuity. The contraction is a polynomial identity in $q$; hence [F24]'s inverse-map calculation still makes the extended metric Lorentzian and jointly smooth. Its time orientation is transported by $A_q$ as in [F3]. This supplies the open analytic interval also at the endpoint without extending any electromagnetic equation. The field and admissible graph neighborhood in step 3.1, the reference estimates just obtained, and step 2.2 meet all the hypotheses of [F19]. It gives the global $C^1$ map between the Banach Dirichlet spaces for every $-1<\beta <0$, using [F20] and [F21]. No Jacobi inverse is asserted.

The graph variation in direction w has velocity wY and normal speed a_*w. Since the reference slice is maximal, its tangential transport term vanishes in [F22]. This gives the first derivative identity. For the constant rescaling $\bar{h}=q_{\ast }^{-2}{h}_{\ast }$, the coordinate determinant-and-inverse formula for the Laplacian gives ${\mathrm{\Delta }}_{\bar{h}}=q_{\ast }^2{\mathrm{\Delta }}_{h_{\ast }}$; substitution gives the second identity. This proves the claimed linearization, weighted estimates and mapping properties under the existing-slice hypotheses.