The global Dirichlet mean-curvature map is continuously differentiable

On a smaller compact parameter interval, positivity of $b,c$ bounds them above and away from zero. The uniform end estimates and $D{u}_{\ast }=O(r^{-7\mathrm{/}4})$ imply bounded ambient components and $h_{q,ij}=g_{q,ij}+g_{q,0i}{u}_{\ast ,j}+g_{q,0j}{u}_{\ast ,i}+g_{q,00}{u}_{\ast ,i}{u}_{\ast ,j}=b(q{)}^2{\delta }_{ij}+O(r^{-1})$ uniformly. Thus $h_q\ge m\delta$ on a sufficiently distant common tail for some $m>0$. These are the end hypotheses of [F2], with stationary generator normalization one. On compact subsets, joint smoothness and reference spacelikeness give its remaining family hypotheses after shrinking the interval. Invoke [F2], carrying [F9], to fix $Y$, its flow domain and an open admissible graph ball. On the tail where $Y=T$ the graph is exactly the time translation in the Statement. Hence [F4] applies with all its reference-height and parameter-norm assumptions supplied explicitly; on compact truncations the fixed smooth flow charts and locally extended smooth data meet [F3]. The spaces and their sum-norm product are Banach by [F1] and [F10].

Apply [F3] on a compact truncation containing the nontranslation region and a fixed overlap collar, and [F4] on the translation tail. Their outputs agree on the overlap because both compute the future geometric mean curvature pulled back by the same $F_w$. Their derivatives agree there as well: at each overlap point subtract the two Fréchet remainder identities along a fixed increment direction, divide by its scalar size and let that scalar tend to zero. The compact and end derivatives therefore define one linear operator $L_{q,w}$ on $\mathbb{R}\times X_{\beta }$. Restrictions of a height to this fixed truncation are bounded in its finite-chart order-four Hölder norm by its global weighted norm: on the added compact end collar only bounded-radius annuli occur, so their weight factors and all fixed chart-transition coefficients have finite bounds. The same finite-chart comparison for order two shows that the global target norm is bounded by a constant times the sum of the compact target norm and the tail norm. Thus, for $\epsilon =\mathrm{\mid }s\mathrm{\mid }+\mathrm{\parallel }v{\mathrm{\parallel }}_{C_{\beta }^{4,\alpha }}$, the global remainder is bounded by $C({\omega }_K(\epsilon )+{\omega }_E(\epsilon ))\epsilon$, where the compact and end moduli tend to zero by [F3] and [F4]. Their derivative bounds also give boundedness of $L_{q,w}$, and the same comparison applied to derivative differences gives operator-norm continuity. By [F8] this proves the asserted global $C^1$ property. Zero trace constrains the height directions only and imposes no output boundary condition.

At fixed $q=q_{\ast }$, the variation $F_{sv}$ has velocity $vY$. Its normal speed is $-g_{q_{\ast }}(vY,n_{\ast })=a_{\ast }v$, and its remaining velocity component is tangent. Apply [F5] with the data and conventions in [F6]. Since $H_{\ast }=0$ identically, tangential transport contributes zero, leaving $({\mathrm{\Delta }}_{h_{\ast }}-V_{\ast })(a_{\ast }v)$. This calculation also applies to every $v\in X_{\beta }$, without assuming density of smooth functions in its Hölder norm: at any interior point the local jet derivative from [F3] or [F4] depends only on the two-jet of $v$, which can be matched by a smooth local polynomial direction; [F5] identifies that same jet expression for smooth local variations. The identity extends to $S$ by continuity of the derivatives up to the boundary. It is consequently the height restriction of the global Fréchet derivative from step 2.1, so is bounded from $X_{\beta }$ to $Y_{\beta }$. Positivity of $a_{\ast }$ alone is not used to infer invertibility.