Theorem 3.1: complete maximal slices for positive q

Adopt the stationary vacuum model and regular data with original maximal exterior section, mass-one normalization and $T{\mathrm{\mid }}_S\equiv ̸0$. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Mass-one convention and Harrison denominator The graph, maximality, induced completeness and Holder conventions are Stationary maximal graphs and normalized graph heights, Spacelike sections, maximality and lapse–shift data and Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains. Work in ZFC, including The Axiom of Choice.

Individual admissibility and estimates for any indexed collection of already existing sections are Uniform end estimates indexed only by existing admissible slices. Compact heights and horizon estimates for exactly those indexed pairs are Indexed compact heights and uniform smooth horizon collars.

Arbitrary convergent existing-candidate sequences have the required compact normal/lapse estimates and admissible complete limiting sections. Compact estimates for arbitrary sequences of individually existing slices Sequence closedness and completeness without an interval-family premise.

The actual existence set is all positive parameters by the individual-reference openness and arbitrary-sequence closedness argument in Open–closed continuation for the set of individually admissible parameters. This replaces any invocation of an interval-family hypothesis to prove interval existence.

The Harrison tensor formula is Stationary Harrison one-form and metric. Under the regular vacuum hypotheses in [A1], Closed Ernst form and existence of normalized potential supplies the normalized smooth stationary Ernst potential $\sigma =f+i\psi$, with $f=-g(T,T)$, real $\psi$ and $\sigma \to 1$, on the exterior and an open neighborhood of $S$. For this supplied potential, the regular vacuum, original maximal outer-section and mass-one hypotheses of Positivity of the Harrison denominator and horizon bound are exactly those in [A1]. It gives $D_q>0$ on the exterior and on one open neighborhood of $S$, independent of $q\in [0,1]$, so the denominators in the Harrison tensor formula are positive. Under exactly the regular vacuum, original maximal-section, mass-one and $T{\mathrm{\mid }}_S\equiv ̸0$ hypotheses in [A1], Proposition 2.1: Einstein–Maxwell equations and regularity constructs the chosen stationary one-form family and jointly smooth Lorentzian metrics $g_q$ on the exterior and a common neighborhood of $S$, with $g_1=g$. Under the same regular vacuum hypotheses, Remark 1.3: lapse bounds and stationary product gives the actual section $C={\mathrm{\Sigma }}_0\cap \mathcal{E}$, with connected smooth spherical-exterior closure ${\mathrm{\Sigma }}_1=C\cup S$ and a physical asymptotically Euclidean end with compact truncations. Original maximality in [A1] is maximality of this actual exterior section, so the theorem's conditional stationary-flow diffeomorphism $\mathbb{R}\times C\to \mathcal{E}$ applies. Its adapted topological radius may put $C$ at $r_{\mathrm{r}\mathrm{e}\mathrm{f}}>1$; this is distinct from the physical end radius $r=\mathrm{\mid }x\mathrm{\mid }$. Thus the metric and product required by the graph definition are supplied before any transformed maximal section is selected.