I'm trying to understand an argument in Casim Abbas' 'An Introduction to Compactness Results in Symplectic Field Theory'. Here's where I stuck: (It's on chapter 3.3.2, Adding additional marked points in the book, if you have access to it.)
Suppose we have a marked Riemann surface $(S,M)$, where $S$ is the Riemann surface and $M\subseteq S$ is the set of marked points. (For simplicity, assume there is no nodal points on $S$.)
Then we are interested in the limit of the sequence of marked Riemann surface $S_n:=(S,M\cup \left\{y_n^{(0)},y_n^{(1)}\right\})$ where the added marked points are given so that $d_n(y^{(0)}_n,y^{(1)}_n)$ goes to zero as $n$ goes to infinity. We further assume that there exists a holomorphic chart of radius $R_n\to \infty$ which maps $y_n^{(i)}$ to $i\in \mathbb{C}$.
Now the argument in the book goes like this:
First, as the two marked points gets closer and closer, we may restrict ourselves on a pair of pants. (In fact, we need to consider the case that the two marked points are lying on two adjacent pairs of pants, but let's just concentrate on this for simplicity.)
So the pair of pants of our concern looks like the first pants as above.
Now he considers a pair of pants decomposition of $S_n$. The following is what was stated:
Because the surface $S_n$ contain annuli $\cong D_{R_n}\setminus D_2$ with larger and larger moduli and $y_n^{(i)}\in D_2$, the last pair of pants decomposition cannot occur by Bers' theorem. In fact, we must have the length of $\gamma_n$ to go to $0$.
Q. Could you explain this part a bit more? I guess that, as we need to use Bers' theorem, it is implicit that the lower dotted circle in the last pair of pants in the figure gets longer and longer as $n$ goes to infinity. Is it correct? Why? And I have no idea how to see the next claim that the length of $\gamma_n$ goes to 0.
Edit: The following is Bers' theorem, paraphrased in my own preference.
(Bers' theorem) Given a topological type (ie, the genus, number of marked points, and the number of boundary components) of a Riemann surface $S$ fixed, there is a uniform bound on the length of each boundary component of each pair of pants in a pair of pants decomposition of $S$ which only depends on the topological type of $S$ and the length of the boundary components of $S$.
Let $M_n$ be the modulus of the annulus $D_{R_n} - D_2$. $M_n \to \infty$ as $n \to \infty$, since $R_n \to \infty$ by assumption. This means that the length of its core curve, say $c_n$, shrinks to $0$. $c_n$ intersects the curve $\gamma_n$ in the last configuration, so by the collar lemma (see, for example the book "Geometry and Spectra for Compact Riemann surfaces", by Peter Buser, for a complete exposition in the case of punctured Riemann surfaces), the length of $\gamma_n$ must be larger than $2M_n$. But this would give a pants decomposition with a boundary curve of unbounded length, contradicting Bers' theorem. This means that only the first three configurations are valid, and in those $\gamma_n$ plays the role of $c_n$, which then must shrink to $0$.