Corollary 4.1: area and pointwise limiting flux rigidity

The stationary vacuum model, regular spacetime, maximal section and mass-one convention are specified by Stationary vacuum model and asymptotic normalization, Regular non-degenerate stationary vacuum spacetime, Spacelike sections, maximality and lapse–shift data and Mass-one convention and Harrison denominator. ZFC includes The Axiom of Choice.

The upper area theorem gives $\mathrm{\mid }S{\mathrm{\mid }}_{g_0}\le 4\pi$; the lower area theorem gives $\mathrm{\mid }S{\mathrm{\mid }}_{g_q}\ge 4\pi (1-q^2)$ for sufficiently small $q>0$. Theorem 4.1: limiting horizon area upper bound Theorem 4.2: charged horizon area lower bound

The regular Harrison result gives a jointly smooth metric family on a neighborhood of the fixed sphere and a two-form family continuous in $q$ and smooth in spacetime. Proposition 2.1: Einstein–Maxwell equations and regularity For $q\in (0,1]$, the spacelike section supplied by the stable-bifurcation result makes ${\gamma }_q$ Riemannian; at $q=0$, the upper-area interface treats $S$ as spacelike, so ${\gamma }_0$ is Riemannian as well. Nonnegative principal eigenvalue of the horizon stability operator Theorem 4.1: limiting horizon area upper bound The horizon and surface conventions define the positive area form and the two density ratios in the Statement. Horizon null frame, flux densities and stability operator notation Surface measure, flux densities and null normal gauge

For each $q\in (0,1]$, the stable-bifurcation result supplies an individually admissible maximal section and a smooth spacelike extension across $S$, in which the compact sphere has an outward normal-graph tube. These are precisely the section, extension and tube hypotheses of the stable-horizon result, which gives ${\int }_S(e_q^2+b_q^2)d{\mu }_q\le 4\pi$ for this $q$. Nonnegative principal eigenvalue of the horizon stability operator Stable-horizon flux-energy and area-charge inequalities

With the stated primitive, end normalization and orientation, the Harrison flux theorem gives ${\int }_S{e}_{q}d{\mu }_q=4\pi \sqrt{1-q^2}$ and ${\int }_S{b}_{q}d{\mu }_q=0$. Proposition 2.1: electric and magnetic fluxes Horizon null frame, flux densities and stability operator notation

On a fixed compact oriented surface, the compact-flux lemma gives convergence of area and of the flux and squared-density integrals under uniform convergence of Riemannian metrics and their associated restricted two-forms. It also proves pointwise $e=1$, $b=0$ when area and electric flux are both $4\pi$ and the squared-density integral is at most $4\pi$. Compact flux saturation and pointwise density rigidity

The Lorentzian Hodge-calculus lemma gives $({\star }_{g}F{)}_{ab}=\frac{1}{2}({\mathrm{v}\mathrm{o}\mathrm{l}}_g{)}_{abcd}{F}^{cd}$ in a positively oriented four-dimensional chart, with both indices of $F$ raised by $g^{-1}$. The volume form uses the metric and the fixed spacetime orientation. Lorentzian exterior calculus and Hodge star