I'm reading a great paper named Geodesically Complete Black Holes by R. Carballo, F. Filippo, S. Liberati and M. Visser about regular black holes and I'm trying to understand a particular situation. To put this into context, I'm interesting in the case named as evanescent horizons that supposes an existence of a set of defocusing points. This particular case is defined as:

$(λ_0, R_0,\theta^{\text{in}}(\lambda_0)<0)$: The expansion $\theta^{\text{out}}$ vanishes and changes sign at a finite affine distance $\lambda=\lambda_0$ or, in terms of the radius, at a value $R_0 > 0$ of the radial coordinate along the congruence of outgoing radial null geodesics at $\lambda=\lambda_0$. On the other hand, the expansion of the intersecting ingoing radial null geodesics remains negative until (and including) $\lambda_0$, so that $\theta^{\text{out}}(\lambda_0)<0$

Where the congruence of outgoing radial null geodesics are all stemming from trapped surfaces (in $\lambda=0$) as said by the authors. So, they states that:

There must exist $2k + 1$ (with $0 \leq k$) open intervals in the domain of $v(R_0)$ where the latter is bijective and the inverse function $R_0(v)$ exists, which must be either a strictly increasing or strictly decreasing function in $k + 1$ of these intervals, and either a strictly decreasing or strictly increasing function in the remaining $k$ intervals.

Before saying what is struggling me, one more result: they also proved that to every constant $v$ there must exist an even number of points $r$ such that $\theta^{\text{out}}$ vanishes.

The basic case happens when there's just $2$ points and its looks standard (the authors already used as an example as Figura 6 shows). My conclusion is that they consider just the red side, instead to consider all the left side, because we are considering only those points related to geodesicas leaving from trapped surfaces. enter image description here

QUESTION: I cannot see how we can have any case with more than $2$ points that $F(v,r)$ vanishes along $v$ in such manner that the above proposition holds.

If someone can help me too see an example, or even knows any other references I'll appreciate very much.


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