Extending a closed one-form potential across a bifurcation sphere

The choice requirements are supplied by [F9]. By [F5], $P$ and $Q$ are open unions of chronological pasts and futures of points. Transitivity from [F6] makes $P$ a past set and $Q$ a future set. If $p\in {\mathrm{\partial }}_{D}P$ and a short smooth future timelike arc $\gamma$ passes through $p=\gamma (0)$, then for $t<0$ clause (c) in [F5] gives $\gamma (t)\in P$. For $t>0$, $\gamma (t)\in \bar{P}$ would imply $p\in I_D^{-}(\gamma (t))\subset P$, contrary to $p\in {\mathrm{\partial }}_{D}P$ and openness of $P$. Thus the positive-time side is outside $\bar{P}$. Reversing time proves that at $p\in {\mathrm{\partial }}_{D}Q$ the positive-time side lies in $Q$ and the negative-time side is outside $\bar{Q}$.

Write $H_P={\mathrm{\partial }}_{D}P$ and $H_Q={\mathrm{\partial }}_{D}Q$ near $S$. By the inline definition their tangent spaces contain no timelike vectors, so $N$ is transverse to each. At any point of either boundary choose local coordinates $(y,t)$ for which ${\mathrm{\partial }}_t$ is future timelike throughout a small box: start with a linear coordinate change making it equal to $N$ at the point, then shrink using the strict timelikeness and orientation inequalities. For a local defining function $h$ of the embedded hypersurface, ${\mathrm{\partial }}_{t}h\mathrm{\ne }0$ there; [F3] applied to $(y,t)\mapsto (y,h(y,t))$ expresses the boundary as a smooth graph $t=a(y)$ after shrinking. Each vertical segment through that graph is an arc of step 1.1. Consequently $r_i=a(y)-t$ is positive exactly on $P$ in a box for $H_P$, and $r_i=t-a(y)$ is positive exactly on $Q$ in a box for $H_Q$. Their zeros are exactly the respective boundaries. Their differentials on $N$ along the boundary are respectively negative and positive, since the kernel is the boundary tangent hyperplane and $N$ and ${\mathrm{\partial }}_t$ lie in the same timelike component.

For each boundary separately, cover $S$ by finitely many smaller coordinate balls whose closures lie in boxes of step 2.1. Choose bumps $b_i$ from [F7] supported in those boxes and equal to one on the smaller balls. Extend each bump and each product $b_i{r}_i$ by zero outside its box; these extensions are smooth because their supports stay inside the box. On the open neighborhood $W=\{{\sum }_i{b}_i>0\}$ of $S$, set $r=({\sum }_i{b}_i{r}_i)\mathrm{/}({\sum }_i{b}_i)$. Every active $r_i$ has the same sign: positive in the chronological set, zero on its boundary, and negative outside its closure. Thus $r$ has exactly those signs and zeros. On the boundary, terms involving derivatives of the weights vanish because $r_i=0$, and $dr(N)$ is a weighted average of strictly negative numbers for $P$, or strictly positive numbers for $Q$; it is nonzero. This constructs smooth defining functions $u$ for $H_P$ and $v$ for $H_Q$ on a common neighborhood of $S$, with $P=\{u>0\}$ and $Q=\{v>0\}$ there. Transversality makes $du,dv$ independent along $S$, with common kernel $TS$. No normal to $\mathrm{\Sigma }$ has been used.

The compact embedded sphere $S$ is closed in the Hausdorff ambient manifold, so [F3] supplies a tubular projection $\pi$ near $S$. For $G=(\pi ,u,v)$, $dG$ is invertible along $S$: $du,dv$ have common kernel $TS$, and $d\pi$ is the identity on $TS$. Shrink the tube so that $G$ is a local diffeomorphism throughout it. It is injective after a further shrinking towards the compact zero section. Indeed, otherwise there are unequal pairs with the same image tending to the zero section. Their equal $\pi$ coordinates and compactness give a subsequence with both points tending to the same point of $S$. Both eventually lie in one inverse chart of [F3], a contradiction. Hence $G$ is a diffeomorphism onto an open neighborhood of $S\times \{(0,0)\}$ in $S\times {\mathbb{R}}^2$. Compactness supplies $\delta >0$ for which this image contains $S\times (-\delta ,\delta {)}^2$.

Let $O=G^{-1}(S\times (-\delta ,\delta {)}^2)$ in that injective tube. This is a connected open neighborhood of $S$. In these coordinates the homotopy $(p,u,v)\mapsto (p,(1-t)u,(1-t)v)$, for $0\le t\le 1$, is a smooth deformation retraction to $S$. By step 3.1 and $E=P\cap Q$, its exterior overlap is exactly $S\times (0,\delta {)}^2$, which is nonempty and connected. This is the same exterior side that meets $C=\chi (\{\mathrm{\mid }x\mathrm{\mid }>1\})$ in [F2].

By [F4] and step 5.1, $H_{\mathrm{d}\mathrm{R}}^1(O)\cong H_{\mathrm{d}\mathrm{R}}^1(S^2)=0$, so $\eta {\mathrm{\mid }}_O=\mathrm{d}{s}_O$ for a smooth real potential; for a complex form apply this to its real and imaginary parts. On the connected open overlap, $\mathrm{d}(s_O-s_E)=0$. By [F8], applied separately to real and imaginary parts if necessary, the difference is one constant. Subtract that constant from $s_O$. The two potentials now agree on their open overlap and therefore define one smooth function on $E\cup O$ with differential $\eta$, extending the specified $s_E$.