Asymptotically flat positive-q ends and cylindrical limit

First verify the one-form hypotheses needed for the component estimates from the supplied primitive. Taking real and imaginary parts in [F8] gives $df(X)=2F(T,X)$ and $d\psi (X)=2({\star }_{g}F)(T,X)$. Since $D_0=(1-f{)}^2+{\psi }^2$, it follows that ${\mathrm{\Theta }}_0(T,X)=2((f-1)df(X)+\psi d\psi (X))=d{D}_0(X)$. In the stationary end chart the coordinate fields commute. With $(W_0{)}_0=W_0(T)=d_0-D_0$, [F9] therefore gives ${\mathrm{\partial }}_t(W_0{)}_i=(d{W}_0)(T,{\mathrm{\partial }}_i)+{\mathrm{\partial }}_i(W_0{)}_0={\mathrm{\partial }}_i{D}_0-{\mathrm{\partial }}_i{D}_0=0$. Also $T(\sigma )=0$ gives ${\mathrm{\partial }}_t(W_0{)}_0=0$. Thus $W_0$ is stationary there. The Killing condition makes the components of $T^{\mathrm{♭}}$ independent of $t$, so the constant-coefficient family in [F8] is stationary for every fixed $q$.

The torsion-free metric-compatible connection in [F8] satisfies ${\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y]$. Expanding [F9] for $T^{\mathrm{♭}}$ consequently gives $(d{T}^{\mathrm{♭}})(X,Y)=g({\mathrm{\nabla }}_{X}T,Y)-g({\mathrm{\nabla }}_{Y}T,X)=F(X,Y)-F(Y,X)=2F(X,Y)$. The same exterior-derivative formula is real-linear in the one-form. For fixed $q$ its coefficients are constant, so $d{W}_q=4(1-q{)}^2((f-1)F+\psi {\star }_{g}F)-8q(1-q)F=4(1-q)(((1-q)f-(1+q))F+(1-q)\psi {\star }_{g}F)$. The contraction identity is already established in [F8]. Together with step 1.1 these are exactly the conditions in [F9]; the corresponding transformed metric in the Statement is the metric specified there. The positivity domain is supplied by [F1]. Thus the one-form hypotheses of the component supplier are established before its estimates are used below, while the given radial choice and section data are retained.

Fix $\delta \in (0,1]$. For $q\in [\delta ,1]$, the factors $a$, $a^{-1}$ and their spatial derivatives have bounds controlled by $\delta$, with each derivative gaining one power of $r^{-1}$. Since $a=q+r^{-1}$, [F2] gives $(g_q{)}_{00}=-(d_q\mathrm{/}(4q){)}^2+O_5(r^{-1})$, $(g_q{)}_{0i}=O_5(r^{-2})$, and $(g_q{)}_{ij}=q^2{\delta }_{ij}+O_5(r^{-1})$, with a common radius and constants depending on $\delta$. For example, $a^{-2}-q^{-2}=-(2q\mathrm{/}r+r^{-2})\mathrm{/}(q^2{a}^2)$ has the required differentiated $O_5(r^{-1})$ bound.

At $q=0$, [F2] gives $(g_0{)}_{ij}=r^{-2}{\delta }_{ij}+O_5(r^{-3})$. The error is smaller than the positive leading form for sufficiently large $r$, so a constant-$t$ section is spacelike there. With $x=r\omega$ and $s=\log r$, $r^{-2}{\delta }_{ij}d{x}^{i}d{x}^j=d{s}^2+{\gamma }_{{\mathbb{S}}^2}$. Indeed $dx=r\omega ds+rd\omega$, and $⟨\omega ,d\omega ⟩=0$ follows by differentiating $\mathrm{\mid }\omega {\mathrm{\mid }}^2=1$, so the leading form is exactly the product with the induced round metric in [F3]. On relatively compact angular charts in a finite cover of the sphere, every required derivative of $x=e^s\omega$ is $O(r)$. Each transformed error component has two coordinate-Jacobian factors. After at most five further derivatives, the chain and product rules express every term as a Cartesian derivative of order $j\le 5$ of the error, bounded by $C{r}^{-3-j}$, times factors bounded by $C{r}^{j+2}$. Thus all these terms are $O(r^{-1})=O(e^{-s})$. This proves the cylindrical metric assertion.

For the additional global-choice conclusion fix $\delta >0$. The regular model in [A1], mass-one data in [A2], and maximality of the specified exterior section and $T{\mathrm{\mid }}_S\equiv ̸0$ in [F10] match the global primitive theorem's hypotheses. Apply it with its supplied pole and normalization in the original chart, as in [F10]. The primitive in F10 and its constant-coefficient family in F8 satisfy the one-form conditions by the local calculations in steps 1.1 and 2.1, which use no radial identity. For $q\in [\delta ,1]$, the Ernst bounds in F10 give $D_q=4q^2+O_5(r^{-1})$, uniformly in $q$. On a common larger end $D_q\ge 2{\delta }^2$. Repeated product and reciprocal differentiation using F5 gives $D_q^{-1}=(4q^2{)}^{-1}+O_5(r^{-1})$. Also $(W_q{)}_i=(1-q{)}^2(W_0{)}_i-4q(1-q)g_{0i}=O_5(r^{-2})$ uniformly. At the fixed pole, $d_q=(1+q{)}^2+(1-q{)}^2\psi (p{)}^2$, so $d_q$ and its reciprocal are bounded on $[\delta ,1]$. Substitute these estimates into the exact identities in F10. They give $(g_q{)}_{00}=-d_q^2\mathrm{/}(16q^2)+O_5(r^{-1})$, $(g_q{)}_{0i}=O_5(r^{-2})$ and $(g_q{)}_{ij}=q^2{\delta }_{ij}+O_5(r^{-1})$ for this same global primitive. Every derivative assertion follows by the finite coordinate product and reciprocal rules; constants may depend on $\delta$.

By [F1], $d_q>0$, so the coordinate change is invertible. Its inverse Jacobian has $dt\mathrm{/}d{t}_q=4q\mathrm{/}{d}_q$ and $\mathrm{\partial }{x}^i\mathrm{/}\mathrm{\partial }{x}_q^j={\delta }_j^i\mathrm{/}q$. Applying the tensor transformation law to step 3.1 gives the displayed leading coefficients. Each $x_q$ derivative contributes a factor $q^{-1}$, and $r=r_q\mathrm{/}q$; all factors are uniformly bounded for $q\in [\delta ,1]$. This proves the stated $O_5$ estimates on a common normalized end. The identity $r_q=qr$ gives the region correspondence and its recession.

Restricting the metric in step 3.2 to $s=\log R$ gives ${\gamma }_{{\mathbb{S}}^2}+O(R^{-1})$ in smooth angular charts. Relative to the positive round metric, the two eigenvalues are $1+O(R^{-1})$, uniformly over the compact sphere. By the determinant density in [F4], the area density is therefore $(1+O(R^{-1})){\mu }_{{\gamma }_{{\mathbb{S}}^2}}$. Integrating and using [F3] gives sphere area $4\pi +O(R^{-1})$, proving the stated limit.

Write $h$ for the zero-$q$ spatial metric and $h_c=d{s}^2+{\gamma }_{{\mathbb{S}}^2}$. Choose $s_0$ so large that on the connected end $E_c=(s_0,\mathrm{\infty })\times {\mathbb{S}}^2$ one has $h_c\mathrm{/}2\le h\le 2h_c$. Fix any $b=(s_b,{\omega }_b)\in E_c$. For every piecewise $C^1$ path from $b$ to $(s,\omega )$ confined to $E_c$, [F4] gives length at least $\mathrm{\mid }s-s_b\mathrm{\mid }\mathrm{/}\sqrt{2}$. Thus its intrinsic ball $B_h^{E_c}(b,\rho )$ lies in $(s_0,s_b+\sqrt{2}\rho )\times {\mathbb{S}}^2$. The volume density is at most $2^{3\mathrm{/}2}{\mu }_{h_c}$ and ${\mu }_{h_c}=ds{\mu }_{{\gamma }_{{\mathbb{S}}^2}}$, so ${\mathrm{vol}}_h{B}_h^{E_c}(b,\rho )\le 2^{3\mathrm{/}2}4\pi (s_b-s_0+\sqrt{2}\rho )$. This proves the linear upper bound without assuming a minimizing curve exists.

Suppose the same infinity admitted the asymptotically Euclidean chart specified in the Statement. Truncate it to an end $E_A=\{\mathrm{\mid }y\mathrm{\mid }>A\}$ lying in $E_c$, with $\delta \mathrm{/}2\le h\le 2\delta$ there. Fix $b^{\mathrm{\prime }}$ with $\mathrm{\mid }y(b^{\mathrm{\prime }})\mathrm{\mid }=2A$. Any $y$ with $2A\le \mathrm{\mid }y\mathrm{\mid }\le L$ can be reached from $b^{\mathrm{\prime }}$ by a spherical arc at radius $2A$ followed by a radial segment, all in $E_A$. By [F3], the Euclidean spherical arc has length at most $2\pi A$; the radial segment has length at most $L$. Its $h$-length is at most $\sqrt{2}(L+2\pi A)$. Consequently this annulus is contained in the intrinsic ball of $E_c$ centered at $b^{\mathrm{\prime }}$ with radius ${\rho }_L=\sqrt{2}(L+2\pi A)+1$, since all these paths are also allowed in $E_c$. For $L>4A$ the annulus contains the Euclidean ball of radius $L\mathrm{/}4$ centered at $(3L\mathrm{/}4,0,0)$ in the $y$ chart. The density lower bound and [F7] give ${\mathrm{vol}}_h{B}_h^{E_c}(b^{\mathrm{\prime }},{\rho }_L)\ge 2^{-3\mathrm{/}2}(4\pi \mathrm{/}3)(L\mathrm{/}4{)}^3$. This is a cubic lower bound for radii ${\rho }_L=O(L)$, contradicting step 4.3, which applies to every fixed center in $E_c$.

Thus there is no asymptotically Euclidean chart at this infinity even in the weaker metric sense specified, and in particular no fixed-chart ADM end of [F6]. Together with steps 4.1, 3.2 and 4.2 this proves all the normalized positive-$q$ estimates, their uniformity and recession, and the cylindrical limit with area $4\pi$. The inherited choice premise is used only through the round-sphere and volume-measure suppliers, not to choose a family of curves in the comparison.

Apply $dt=(4q\mathrm{/}{d}_q)d{t}_q$ and $d{x}^i=d{x}_q^i\mathrm{/}q$ to the bounds in step 3.3. The tensor transformation law in F5 and $r=r_q\mathrm{/}q$ give exactly the stated normalized $O_5$ bounds, uniformly on a common end for $q\in [\delta ,1]$. The graph of the coordinate change has $r_q=qr$, so the same region correspondence and recession follow. This proves the additional positive-parameter conclusion for the actual smooth global family while preserving all radial-choice conclusions already proved. No zero-parameter assertion for a different gauge is inferred from this extension.