Negative stability eigenvalue produces outer trapped graphs

PDF p. 27: Proof of Theorem 4.2, contradiction setup

Adopt the regular stationary vacuum model, original maximal exterior section, mass-one normalization and nontriviality of $T{\mathrm{\mid }}_S$ in Horizon expansion variation and Einstein–Maxwell curvature. Fix $q\in (0,1]$ and supply the smooth spacelike maximal section ${\mathrm{\Sigma }}_q$ with boundary $S$, its smooth spacelike extension across $S$, outward unit normal ${\nu }_q$, and admissible normal-graph tube required there. The bifurcation sphere $S$ is nonempty, connected and compact without boundary. Write $h_q$ for the induced Riemannian metric on the extended section, $n_q$ for its future unit normal, and ${\mathcal{L}}_q$ for the outward stability operator in the fixed gauge $L_q=n_q+{\nu }_q$ of Horizon null frame, flux densities and stability operator notation.

If the principal eigenvalue ${\lambda }_1$ of ${\mathcal{L}}_q$ is negative, there is a smooth everywhere positive eigenfunction $\varphi$ such that, for all sufficiently small $ϵ>0$, the outward normal graph $S_{ϵ}=\{{\exp }_x^{h_q}(ϵ\varphi (x){\nu }_q(x)):x\in S\}$ has strictly negative outgoing expansion in the gauge $n_q+{\nu }_{q,ϵ}$, where ${\nu }_{q,ϵ}$ is its outward unit normal in the extended section. Thus each such graph is outer trapped: its outgoing expansion is negative at every point. After fixing one such $ϵ$, there is ${\delta }_{\ast }>0$ such that ${\theta }_{n_q+{\nu }_{q,ϵ}}(S_{ϵ})\le -{\delta }_{\ast }$ everywhere. This ${\delta }_{\ast }$ is local to the fixed graph and is unrelated to the uniform parameter interval in Section 3.

The article reference attributes the contradiction setup. The proof uses the stated completed suppliers and does not take the article's claim as a premise.