Stationarity supplies the weighted end differentiability estimates

Shrink $J$ about $q_{\ast }$ so positivity and continuity of $b,c$ bound them above and away from zero. The common decay bounds and $D{u}_{\ast }=O(r^{-7\mathrm{/}4})$ give, on a sufficiently distant common tail, bounded ambient components and $h_{q,ij}=b(q{)}^2{\delta }_{ij}+O(r^{-1})$ for the reference graph. Thus $h_q\ge mI$ for a common $m>0$. The hypotheses of [F5] hold there and on the compact part after shrinking the parameter interval about $q_{\ast }$. Fix its $Y$ and increase the tail radius until $Y=T$. On this tail every height graph has unchanged spatial coordinates and is an embedding in the supplied product, whether or not its boundary trace vanishes. The exact induced-metric formula in [F5] preserves $h_{q,w}\ge (m\mathrm{/}2)I$ for uniformly small $\mathrm{\mid }Dw\mathrm{\mid }$, controlled by the weighted norm since $\beta <0$. The metric arrays, their inverses and all slopes $D{u}_{\ast }+Dw$ consequently lie in bounded ranges uniformly separated from degenerate metrics and null graph conormals. This supplies the common admissible coefficient range; it is not inferred from compact-core coefficient bounds.

In the stationary coordinates write $s=D{u}_{\ast }+Dw$ and $s_{\ast }=D{u}_{\ast }$. Applying [F4] directly to the graph $t=u_{\ast }+w$ gives ${\mathcal{H}}_E(q,w)=A^{ij}(g_q,s)(u_{\ast ,ij}+w_{ij})+B(g_q,{\mathrm{\partial }}_x{g}_q,s)$. Here $A$ is a smooth function of the metric array and slope on the range from step 1.1, and $B$ is a finite sum of a first spatial metric derivative times a smooth function of that array and slope. Indeed the conormal is $dt-s_{i}d{x}^i$; its normalized metric dual depends algebraically only on $(g_q,s)$, and its tangential derivative consists of terms linear in ${\mathrm{\partial }}_x{g}_q$ and in $D^2(u_{\ast }+w)$. The connection terms are also linear in ${\mathrm{\partial }}_x{g}_q$. Stationarity removes time derivatives of the metric and every dependence on the value of $u_{\ast }+w$. All signs are fixed by [F8].

Regard all the following coefficients as $C^{2,\alpha }$ arrays on the end. The metric array is a $C^1$ function of $q$ into weight zero, while ${\mathrm{\partial }}_x{g}_q$ is $C^1$ into weight $-2$, because its spatially constant leading terms differentiate to zero. Also $s_{\ast }\in C_{-7\mathrm{/}4}^{2,\alpha }$ and $D^2{u}_{\ast }\in C_{-11\mathrm{/}4}^{2,\alpha }\subset C_{-2}^{2,\alpha }$. The maps $w\mapsto Dw$ and $w\mapsto D^{2}w$ are bounded into weights $\beta -1$ and $\beta -2$, respectively, and these include into the zero-weight space. By [F3] on the common range in step 1.1, $A(q,w)=A(g_q,s_{\ast }+Dw)$ is $C^1$ with values in weight zero. Every smooth coefficient multiplying ${\mathrm{\partial }}_x{g}_q$ in $B$ has the same property, as do its finite slope derivatives. The statements include full finite-norm Hölder inputs, not just smooth perturbations.

Subtract the reference expression from step 2.1. Smooth finite-dimensional divided differences give $A(g,s_{\ast }+v)-A(g,s_{\ast })=A_1(g,s_{\ast },v)v$ and $B(g,d,s_{\ast }+v)-B(g,d,s_{\ast })=B_1(g,d,s_{\ast },v)v$, where $A_1$ and $B_1$ are smooth on a slightly smaller common range, and $B_1$ is linear in $d$. For example these divided differences are the componentwise integrals of the slope derivatives from slope $s_{\ast }$ to slope $s_{\ast }+v$; the slope segment remains admissible by the small-slope bounds of step 1.1. Consequently ${\mathcal{H}}_E(q,w)=h(q)+A^{ij}(q,w)w_{ij}+C^j(q,w)w_j$, where $C$ is obtained by contracting $A_1(g_q,s_{\ast },Dw)$ with $D^2{u}_{\ast }$ and adding $B_1(g_q,{\mathrm{\partial }}_x{g}_q,s_{\ast },Dw)$. Step 3.1 and [F2]–[F3] show that $C(q,w)$ is $C^1$ with values in weight $-2$. Their finite-dimensional smoothness follows by differentiating the integrands on a slightly smaller coefficient range: continuity of each derivative on the compact integration interval gives uniform convergence of the corresponding difference quotients, as in [F3], Proof 4.1. No integration in a function space is assumed.

The product $A(q,w):D^{2}w$ is $C^1$ into weight $\beta -2$ by the bounded bilinear multiplication and composition rules of [F2]–[F3]. The product $C(q,w)\cdot Dw$ is $C^1$ into weight $\beta -3$, which includes continuously into weight $\beta -2$. Finally $h(q)$ is $C^1$ into weight $-3$ by [F1], and $-3<\beta -2$ because $\beta >-1$. Their sum is the asserted $C^1$ map on a product neighborhood, hence on a small ball about $(q_{\ast },0)$. Explicitly its height derivative is $D_w{\mathcal{H}}_E(q,w)v=A(q,w):D^{2}v+C(q,w)\cdot Dv+(D_{w}A(q,w)v):D^{2}w+(D_{w}C(q,w)v)\cdot Dw$; its parameter derivative is $h^{\mathrm{\prime }}(q)+({\mathrm{\partial }}_{q}A(q,w)):D^{2}w+({\mathrm{\partial }}_{q}C(q,w))\cdot Dw$. The displayed products prove boundedness in the target norm.

The remainder and derivative-continuity estimates in [F3] use one common coefficient range from step 1.1, the parameter continuity in the assumed weighted norms, and the bounded linear and bilinear operations in steps 3.1–4.1. They therefore have a single modulus tending to zero in the sum norm $\mathrm{\mid }\delta q\mathrm{\mid }+\mathrm{\parallel }\delta w{\mathrm{\parallel }}_{C_{\beta }^{4,\alpha }}$, before taking the supremum over exterior annuli. The bounded product rules preserve that modulus in the two displayed derivative formulas, proving operator-norm continuity and a uniform Fréchet remainder. Equivalently, on $x=Ry$ the normalized perturbation $\eta =R^{-\beta }w(Ry)$ has physical slope $R^{\beta -1}D\eta$ and Hessian $R^{\beta -2}{D}^2\eta$; the $C\cdot Dw$ term has an extra bounded factor $R^{-1}$ in the normalized target, and no term arises from undifferentiated height. On any fixed end collar the same smooth coefficient formulas and [F2]–[F4] give the ordinary $C^{4,\alpha }$ to $C^{2,\alpha }$ estimate, so that collar can be included. Restriction to the closed zero-trace subspace preserves continuous differentiability and agrees with the global admissible graphs in [F5]. This proves all the asserted end estimates.