Outer RN escape boundary in selected model

On a compact path segment contained in the collar, write $w=r_H-r$ while $r\le r_H$. From [F2], on each $C^1$ piece, $w^{\mathrm{\prime }}=-dr(Z)\ge -Cw{\tau }^{\mathrm{\prime }}\mathrm{.}$ By [F3], $(e^{C\tau }w{)}^{\mathrm{\prime }}\ge 0$. The factor is positive and continuous across the finite partition, so $w$ cannot decrease to zero from a positive value while the path remains below the root in the collar. Thus no path starting below $r_H$ can reach the root: if it did, continuity gives a final collar segment starting either at its initial point inside the collar or at the last crossing of a fixed smaller collar lower boundary, and ending at the first root contact. Its starting value of $w$ is positive, contradicting this inequality. For a path starting at the root, suppose $r>r_H$ somewhere in a compact segment. A connected component of the open parameter set $\{r>r_H\}$ then has a left endpoint at which $r=r_H$. Restrict to an initial portion of that component lying inside the collar, stopping at its first crossing of $r_H+\delta \mathrm{/}2$ if necessary. On this collar portion let $h=e^{-C\tau }(r-r_H)$. By [F2-F3], $h^{\mathrm{\prime }}\le 0$ on every $C^1$ piece, so [F4] makes $h$ nonincreasing across the finite partition. Continuity at the left endpoint gives $h\le 0$ throughout the component, contradicting $r>r_H$. Thus a path from the root cannot make a positive excursion. In particular neither a point below nor a point on the root can escape to $r>r_H$.

Fix $r_0>r_H$, $v_0\in \mathbb{R}$, and ${\omega }_0\in S^2$. For $r\ge r_0$, set $v(r)=v_0+{\int }_{r_0}^r\frac{2}{f(s)}ds\mathrm{.}$ Here $f(s)=(s-r_{+})(s-r_{-})\mathrm{/}{s}^2>0$ because $s>r_{+}$, so [F4] gives $v^{\mathrm{\prime }}(r)=2\mathrm{/}f(r)$. The curve $\gamma (r)=(v(r),r,{\omega }_0)$ has tangent ${\gamma }^{\mathrm{\prime }}(r)=\frac{2}{f(r)}{\mathrm{\partial }}_v+{\mathrm{\partial }}_r=\frac{2}{f(r)}n,$ a positive multiple of the future null field in [F1]. Reparameterize by $s=r-r_0\in [0,\mathrm{\infty })$ to obtain the path domain in [L1]. It is defined for every finite $r\ge r_0$ and therefore reaches arbitrarily large radii in the exact path sense of [L1]. Every point with $r>r_H$ lies in $\mathcal{E}$.

Steps 1.1 and 1.2 show respectively that no point with $r\le r_H$ escapes and every point with $r>r_H$ escapes. The boundary of this open radial half-space in ${\mathbb{R}}_v\times (0,\mathrm{\infty }{)}_r\times S^2$ is the level set $r=r_H$. This proves the claim under the conditional escape definition [L1].