Nonnegative principal eigenvalue of the horizon stability operator

Fix $q\in (0,1]$. The hypotheses of [F0] are exactly [A1], so use its single primitive, positive denominator, regular metric and stationary product throughout. By [F1], choose an individually admissible section. Apply its sequence-closure result on $[q\mathrm{/}2,1]$ to the constant sequence consisting of this same section and parameter. A constant sequence of the stationary graph functions and boundary graph representations has that same section as its limit. Hence the chosen section is complete with $S$ included.

Verify the invariant construction for these actual fields. The regular model, original maximality and $T{\mathrm{\mid }}_S\equiv ̸0$ in [A1] meet the local-field hypotheses of [F8]; use the $K,Z$ and common pole $p$ employed by the global construction in [F0]. The invariant-data result applies with the same mass-one convention and normalized Ernst potential, so that potential is locally $K,Z$ invariant. The already chosen global $W_0$ is locally $K,Z$ invariant by [F8]; no replacement primitive is needed. Hence the very family $W_q=(1-q{)}^2{W}_0-4q(1-q)T^{\mathrm{♭}}$ in the Statement is the invariant horizon family: the commuting Killing fields preserve $T^{\mathrm{♭}}$, as in the invariant-data result. The horizon-preservation hypotheses are therefore met by this family and [A1]. Define $F_q$ by [F9] from these same data and use the electrovacuum realization in [F0]. Intersect its open neighborhood of $S$ with the invariant-data and horizon-preservation neighborhoods, obtaining an open $O$ containing $S$. On $O$, both applications use the identical metric formula, potential, pole and primitive. The real two-form is the single $F_q$ just defined; the horizon-preservation result concerns this metric and imposes no competing electromagnetic choice. Retain the original spacetime orientation, the electrovacuum time orientation $A_q^{-1}N$, and the orientation of $S$ transported from outward end spheres. Thus there is one oriented, time-oriented smooth pair $g_q,F_q$ on $O$, satisfying electrovacuum and horizon preservation for the supplied construction.

We construct a smooth spacelike extension using one ambient tubular chart. Take a collar $c:S\times [0,a)\to {\mathrm{\Sigma }}_q$ from [F7], with $c(s,0)=s$. Its inward transverse vector at $S$, projected orthogonally off $TS$ within the positive metric on $T{\mathrm{\Sigma }}_q$ and normalized, gives a smooth inward unit normal $\mu$ to $S$ within the section. Together with the smooth future unit normal $n_q$, it gives the normal-quotient isomorphism ${\alpha }_s(r,z)=[r{\mu }_s+z{n}_q(s)]$ from $S\times {\mathbb{R}}^2$ to $TM{\mathrm{\mid }}_S\mathrm{/}TS$. Work inside the common open neighborhood $O$ from step 2.1, with its smooth metric $g_q$. The sphere is compact and therefore closed in this ambient manifold. Apply [F7] to obtain a diffeomorphism $\mathrm{\Phi }$ from a neighborhood of $S\times \{(0,0)\}$ onto an ambient neighborhood of $S$, fixed on $S$, inducing $\alpha$. Shrinking using compactness gives a uniform fiber box. Write its coordinates as $(s,r,z)$, and let $P(\mathrm{\Phi }(s,r,z))=(s,r)$. Since $d\mathrm{\Phi }({\mathrm{\partial }}_r)-\mu$ and $d\mathrm{\Phi }({\mathrm{\partial }}_z)-n_q$ lie in $TS$ along the zero section, $dP{\mathrm{\mid }}_{T{\mathrm{\Sigma }}_q}$ is invertible there, and its $r$ coordinate increases on inward transverse vectors.

Here is the boundary and global compatibility check for that projection. In collar coordinates $(y,t)$ the map $P\circ c$ fixes $(y,0)$, and its last coordinate $R(y,t)$ has $R(y,0)=0$ and ${\mathrm{\partial }}_{t}R(y,0)>0$. Extend its coordinate functions across $t=0$ by [F2]. The derivative remains invertible there, so the Euclidean inverse theorem in [F7] gives local smooth inverses. After shrinking, the smooth extension has ${\mathrm{\partial }}_{t}R>0$. For each fixed $y$, apply the real mean value theorem in [F7] on the interval between $0$ and $t$: smoothness supplies continuity on that closed interval and differentiability inside it, and $R(y,0)=0$. Thus $R$ and $t$ have the same sign, so the local inverses restrict to inverses of the half-collars. Shrink the collar into these local inverse domains. They give local scalar representations $z=v(s,r)$ for $r\ge 0$. The projection is injective on some smaller whole collar: otherwise distinct pairs in collars of widths tending to zero would have equal projections; compactness gives limits on $S$, equality of projections gives the same limiting point, and both pairs eventually lie in one of the injective local inverse neighborhoods just obtained. Its image is relatively open around the compact seam, so contains $S\times [0,b)$ for some $b>0$. Restricting to that cylinder therefore gives one smooth real function $v$, rather than graph functions in different ambient coordinates.

Apply [F2] to this scalar $v$ on $S\times [0,b)$, obtaining smooth local extensions $v_i$ on open subsets of the signed product $S\times (-b,b)$ covering the seam. Restrict to a smaller signed cylinder inside their union, choose its subordinate smooth partition $({\chi }_i)$, and set $\bar{v}={\sum }_i{\chi }_i{v}_i$. Every $v_i$ equals the same $v$ on its original half-domain, so $\bar{v}=v$ for $r\ge 0$; all derivatives along $r=0$ also agree. Shrink the cylinder so that $(s,r,\bar{v}(s,r))$ remains in the domain of $\mathrm{\Phi }$. Its graph under $\mathrm{\Phi }$ is an embedded smooth hypersurface with $S$ in its interior, agreeing with the original section for $r\ge 0$. At the seam its tangent metric is the original positive-definite metric; openness of positive definiteness and compactness of $S$ make the graph spacelike on a smaller signed cylinder. Finally take the ambient neighborhood of $S$ small enough to meet the original embedded section only in the collar just represented. The negative-$r$ graph then has no intersection with the rest of the section, and gluing on the shared positive collar gives the required smooth spacelike extension of the section. No maximality beyond the original side is claimed.

The outward unit normal ${\nu }_q$ to $S$ in this extension is supplied by the original side and its metric, which the extension preserves. The compact normal-graph result in [F2] now applies with $S$ in the interior of the extended spacelike hypersurface, equipped with the smooth Riemannian metric ${\widehat{h}}_q=g_q{\mathrm{\mid }}_{T{\widehat{\mathrm{\Sigma }}}_q}$. It gives a radius ${\rho }_q>0$ and the asserted diffeomorphism $(p,r)\mapsto {\exp }_p^{{\widehat{h}}_q}(r{\nu }_q(p))$ on $S\times (-{\rho }_q,{\rho }_q)$. Its countable-choice hypothesis is supplied by ZFC; no radius uniform in $q$ or in the choice of section is claimed. Thus the original complete maximal stationary graph section has exactly the extension and graph data required by [F3], [F5] and [F6].

Supply the matching boundary fields required by [F3]. Set $U_S=O_S=O$ and take the metric and two-form there to be restrictions of the single pair in step 2.1. Their agreement with the exterior Harrison fields is literal restriction of the same smooth tensors on the exterior union $O$, with the same spacetime and time orientations. The extension in step 5.1 was constructed in $O$. Shrink its collar and the tube from step 6.1 so that their images lie in $O$; compactness of $S$ permits this because the graph map is continuous and its zero section lies in $O$. The outward unit normal is the one determined by the original exterior side, and the future unit normal uses the unchanged electrovacuum time orientation. These are exactly the boundary data of [F9] in the $n_q+\nu$ gauge. Together with step 2.1 they meet the common invariant-construction, electromagnetic-field and orientation hypotheses of [F3], rather than merely postulating compatible extensions.

The sphere is connected and compact by [A1]. The variation result [F3] gives zero outgoing expansion and the stated smooth elliptic operator in the future/outward gauge. Hence $S$ is a MOTS, and [F4] supplies the real principal eigenvalue and its positive eigenfunction.

Assume for contradiction that ${\lambda }_1({\mathcal{L}}_q)<0$. By [F5], choose a sufficiently small outward eigenfunction graph with outgoing expansion at most $-{\delta }_{\ast }$ for some fixed ${\delta }_{\ast }>0$. The original section is complete, smooth and maximal, its interior is a stationary graph, and it has the required boundary extension and tube by steps 1.1 and 6.1.

Therefore [F6] applies to precisely this graph and gives a focal point before a first-contact target while the same null segment has no such interior focal point. This is a contradiction, so the assumption of a negative principal eigenvalue is false. Consequently ${\lambda }_1({\mathcal{L}}_q)\ge 0$. For the positive principal eigenfunction $\varphi$ from step 8.1, ${\mathcal{L}}_q\varphi ={\lambda }_1({\mathcal{L}}_q)\varphi \ge 0$ everywhere. Thus $\varphi$ is a smooth nonnegative normal speed, not identically zero, with nonnegative first variation everywhere, and [F4] gives the stated outward stability. Since $q$ was arbitrary in $(0,1]$, the sufficiently-small-positive-$q$ claim follows.