Complete volume preserving shift flow on a spherical exterior

Use the fixed global exterior diffeomorphism $F$ from the Statement. For $u\in {\mathbb{S}}^2$ and $0\le \rho \le \epsilon$, the map $\kappa (u,\rho )=F^{-1}((1+\rho )u)$ is a smooth product parametrization of a compact collar of $S$; its inverse is $p\mapsto (F(p)\mathrm{/}\mathrm{\mid }F(p)\mathrm{\mid },\mathrm{\mid }F(p)\mathrm{\mid }-1)$. This constructs the collar directly from the given diffeomorphism. Tangency gives $Z\rho (u,0)=0$, and smoothness and compactness bound $\mathrm{\mid }{\mathrm{\partial }}_{\rho }(Z\rho )\mathrm{\mid }\le K$ on a smaller compact collar. Hence $\mathrm{\mid }Z\rho (u,\rho )\mathrm{\mid }\le K\rho$ by integrating in $\rho$. Along a maximal interior integral curve, $\mathrm{\mid }d(\log \rho )\mathrm{/}ds\mathrm{\mid }\le K$ whenever it lies in that collar. On a time interval of finite length $T$, each visit begins either at the initial point or at the collar edge $\rho =\epsilon$, so this inequality gives a common positive lower bound for $\rho$ throughout all such visits, in either time direction. In the specified end chart, $Zr=-{\sum }_{i=1}^3{X}^i{x}^i\mathrm{/}r$, so $\mathrm{\mid }Zr\mathrm{\mid }\le 3A{r}^{-2}$ for $r\ge R_1$. Choose $R>R_1$; this gives a bound $M$ on $\mathrm{\mid }Zr\mathrm{\mid }$ for $r\ge R$. Integrating from the initial time or the last crossing of $r=R$ bounds $r$ on every finite time interval, in either time direction. Thus the segment lies in some compact $K_{R_{\ast }}$. Removing the open boundary collar below the positive lower bound for $\rho$ leaves a compact subset of $C$ containing that segment. The collar estimate uses $F$, and the end estimate uses $x$; compactness of $K_{R_{\ast }}$ combines their bounds without requiring the two charts to agree.

If a maximal orbit had a finite time endpoint, step 1.1 would confine its finite-time terminal portion to a compact subset of $C$. Cover this compact set by finitely many local-flow neighborhoods, and take the minimum of their positive existence times after shrinking the neighborhoods. At a time sufficiently near the purported endpoint the orbit therefore extends beyond it, contradicting maximality and uniqueness. This applies forward and backward, proving completeness. Local smooth dependence, composed over finitely many local-flow intervals around any fixed finite time, makes $(s,p)\mapsto {\psi }_s(p)$ smooth. Uniqueness gives ${\psi }_{s+t}={\psi }_s\circ {\psi }_t$ and inverse ${\psi }_{-s}$, so every ${\psi }_s$ is a smooth diffeomorphism.

In the global exterior coordinates $y=F(p)$ let $m$ be the Riemannian density coefficient and put $J_s(y)=\det D{\psi }_s(y)>0$, the sign being positive by continuity from the identity and invertibility from step 2.1. Differentiating the flow equation gives ${\mathrm{\partial }}_{s}D{\psi }_s=DZ({\psi }_s)D{\psi }_s$. Jacobi's formula and the chain rule therefore give ${\mathrm{\partial }}_s[m({\psi }_s(y))J_s(y)]=({\mathrm{\partial }}_i(m{Z}^i))({\psi }_s(y))J_s(y)=0$, since ${\mathrm{div}}_{h}Z=-{\mathrm{div}}_{h}X=0$. Hence $m({\psi }_s(y))J_s(y)=m(y)$. For any Borel $A\subset C$, the set ${\psi }_s(A)$ is Borel because ${\psi }_s$ is a homeomorphism. Apply the nonnegative measurable change-of-variables theorem to ${\mathbf{1}}_{{\psi }_s(A)}m$. It gives ${\mu }_h({\psi }_s(A))={\mu }_h(A)$, allowing infinite values. Countable choice required by the measure and change-of-variables suppliers is supplied by the ambient ZFC assumption in [F4].

In any small coordinate ball the positive continuous density has a positive lower bound on a smaller ball; thus every nonempty open set has positive measure. Smoothness and positive definiteness up to compact $S$, together with compactness of the closed bounded annulus in $\bar{C}$, give a bounded smooth density in finitely many bounded charts there. Integrating these bounded densities over their bounded coordinate domains bounds the volume of the interior annulus by a finite sum of finite numbers. Every Borel subset of that interior annulus has no larger measure, even when its closure meets $S$.