Compact flux saturation and pointwise density rigidity

For the rigidity assertion $S$ is nonempty because its stated total mass is $4\pi$. By [F7], each continuous real function on $S$ is bounded. More generally, for any finite Borel measure $\nu$ on $S$ (or its completion) and bounded continuous real $f$, [F8] makes $f$ and $\mathrm{\mid }f\mathrm{\mid }$ measurable. Choose a finite $C>0$ bounding $\mathrm{\mid }f\mathrm{\mid }$; [F3] and [F9] give ${\int }_S\mathrm{\mid }f\mathrm{\mid }d\nu \le {\int }_{S}Cd\nu =C\nu (S)<\mathrm{\infty }$, hence $f\in L^1(\nu )$. Apply this to the given smooth positive measure $\mu$, whose local densities specify its Borel measure, with $\mu (S)={\int }_{S}1d\mu =4\pi$. Put $h=(e-1{)}^2+b^2$. All functions in this expansion are continuous, bounded and integrable by the preceding argument. Expanding the square and applying linearity gives ${\int }_{S}hd\mu ={\int }_S(e^2+b^2)d\mu -2{\int }_{S}ed\mu +{\int }_{S}1d\mu \le 0$. Nonnegativity gives the reverse inequality, so this integral is zero.

In a positive chart, write ${\alpha }_j=a_{j}d{x}^1\wedge d{x}^2$ and ${\beta }_j=c_{j}d{x}^1\wedge d{x}^2$ using [F1], and put $v_j=\sqrt{\det q_j}>0$. Then $e_j=a_j\mathrm{/}{v}_j$ and $b_j=c_j\mathrm{/}{v}_j$ satisfy the required two-form equalities by [F2]; the same formulas define $e,b$ with $v=\sqrt{\det q}$. These smooth local functions agree on overlaps: at each point each is the unique scalar multiplying the same nonzero intrinsic area form to give the specified two-form. Thus the quotients define unique global smooth real functions for arbitrary two-forms on $S$, without an ambient spacetime.

If $h(p)>0$ at some point, continuity gives a coordinate neighborhood on which $h\ge c>0$. Its coordinate image contains a closed rectangle $Q$ with finite endpoints and strictly positive side lengths, so $0<{\lambda }_2(Q)<\mathrm{\infty }$ by [F12]. By [F7], $Q$ is compact and the smooth strictly positive chart density ${\rho }_x$ of $\mu$ has a positive minimum $m$ on $Q$. The chart preimage $R$ of $Q$ is closed in the open chart domain $U$; thus $R=U\setminus V$ for some open $V$ in $S$, making $R$ Borel by [F8]. By [F11], [F3] and [F9], $\mu (R)={\int }_Q{\rho }_{x}d{\lambda }_2\ge m{\lambda }_2(Q)>0$. Monotonicity gives ${\int }_{S}hd\mu \ge c\mu (R)>0$, a contradiction. Hence $h=0$ everywhere, and its two nonnegative square terms give $e=1$ and $b=0$ pointwise.

If $S$ is empty all convergence assertions are immediate; otherwise, around each point choose a positive chart and an open coordinate rectangle whose closed rectangle is contained in the chart image. These open patches cover $S$, so [F4] gives finitely many of them. Each closed rectangle is compact by [F7], and its inverse chart image is compact by the continuous-image assertion of [F7]. On each closed rectangle, the continuous positive $v$ has a positive minimum by [F7]; all limiting metric and form coefficients are bounded there by the same fact. Uniform convergence of the metric coefficients implies uniform convergence of their determinants, hence $v_j\to v$ uniformly, with $v_j$ eventually bounded away from zero. The assumed coefficient convergence and the formulas of step 1.2 therefore give $e_j\to e$ and $b_j\to b$ uniformly on each patch and then on $S$. The global ratio $r_j=v_j\mathrm{/}v=d{\mu }_{q_j}\mathrm{/}d{\mu }_q$ converges uniformly to $1$. Consequently $r_j$, $e_j{r}_j$, $b_j{r}_j$, and $(e_j^2+b_j^2)r_j$ converge uniformly to $1$, $e$, $b$, and $e^2+b^2$, respectively.

The fixed measure ${\mu }_q(S)$ is finite: on each closed coordinate rectangle $Q$ of the finite cover in step 2.2, [F7] bounds the local density $v$ above by a finite $M>0$. Its inverse chart image $R$ is Borel as in step 2.1, and [F11], [F12], [F3] and [F9] give ${\mu }_q(R)={\int }_{Q}vd{\lambda }_2\le M{\lambda }_2(Q)<\mathrm{\infty }$. These finitely many sets cover $S$, so finite subadditivity in [F13] gives ${\mu }_q(S)<\mathrm{\infty }$. The functions in the four global pairs in step 2.2 and their differences are continuous, hence bounded and Borel measurable by [F7] and [F8]. The finite-measure argument of step 1.1 makes them integrable for ${\mu }_q$ and, likewise, for each ${\mu }_{q_j}$, whose total mass is finite by the same finite-cover argument. The coordinate identity $v_j=r_{j}v$ gives the measure identity $d{\mu }_{q_j}=r_{j}d{\mu }_q$: on each Borel subset of a chart, [F11] and [F10] identify both measures with the integral of $r_{j}v=v_j$ against ${\lambda }_2$, and a finite disjoint Borel refinement of the chart cover gives the identity globally. For each pair $u_j,u$ among the four uniformly convergent pairs in step 2.2, linearity, the integral triangle inequality and monotonicity give $\mathrm{\mid }{\int }_S{u}_{j}d{\mu }_q-{\int }_{S}ud{\mu }_q\mathrm{\mid }\le {\mu }_q(S){\sup }_S\mathrm{\mid }{u}_j-u\mathrm{\mid }\to 0$. Applying [F10] to the nonnegative integrands, and separately to the positive and negative parts of the signed integrands using [F9], rewrites these integrals as area, the two flux integrals and the squared-density integral for ${\mu }_{q_j}$. This proves all claimed integral limits.

No derivative of a two-form coefficient was used in steps 1.2–3.1. For continuous two-forms, meaning alternating covector fields with continuous coefficients in smooth coordinate frames, the pointwise basis in [F6] gives unique coefficients $a_j,c_j,a,c$; these are continuous by the stated meaning of continuity in a coordinate frame. Division by the smooth positive $v_j,v$ gives continuous density quotients, and uniqueness again makes them agree on overlaps. Uniform coefficient convergence therefore gives exactly the same uniform bounds and integral limits. In particular the claimed smooth-metric and continuous-two-form convergence case follows.