I need some clarification about the reason why we have a sphere bubbling off in the situation described by Seidel in [The Symplectic Floer Homology of a Dehn Twist][1].

I’ll try to summarize to the best of my abilities the situation I’m interested in:

We are stretching the neck along a circle in $\Sigma$. Let us fix an identification of the tubular neighborhood of such circle with $[-1,1]\times S^1$. Let $R>0$, by $\Sigma^R$ I’ll denote the surface diffeomorphic to $\Sigma$ but with neck $[-R,R]\times S^1$. 

Assume that for $R_i\to \infty$, we have a sequence of $J^{R_i}$-holomorphic strips $\{u^i\}_i$, with $u^i\in \mathcal{M}^{R_i}(x_-,x_+)$ (for a definition see bottom of page 832 of the paper) with uniform bounded energy. 

Notice that we **can’t** have uniform bound on $|du^i|_{L^{\infty}}$, in particular we can find a sequence of points $z_i \in [-R_i,R_i]\times S^1$ such that $|du^i|$ has a maximum there, with value $C_i$, and $C_i\to \infty$. Let $\Sigma’=\Sigma \setminus [-1,1]\times S^1$, (the definition of $\Sigma’$ is independent of the length of the neck!) now assume that, for $i$ big enough, $$d(u^i(z_i),\Sigma’)\leq k < \infty$$

(I.e. no matter the length of the neck that is stretching to infinity, my Maximums stays within finite distance to some edge of the neck)

>From there the author immediately concludes that after some reparametrization, the $u^i$’s converged to a $J$-holomorphic sphere in the surface obtained from $\Sigma’$ by attaching a semi-infinite cylinder to each boundary component. I can show that, after reparametrization my curves converges to a non-constant $J$-hol map defined on $\Bbb C$, but I don’t see why this map should extend to the sphere. My understanding is that in order to use the removable of singularity theorem we need a compact image, but $\Sigma’$ With the cylindrical ends is non compact.

Most likely there is some clever way to reparametrize this curves (i used the standard conformal map $z\mapsto z/C_i+z_i$ that centers $u^i$ at $0$ and normalize the norm of its differential at $0$), or to infer something about the image of the limit, but I’m unable to see it.

Any hint is really appreciated, since this is bugging me. Thanks in advance!

**Update**

I agree with the given answer that *monotonicity lemma* could be a way to establish compactness of the image of the limiting function. What I don't understand are the following point:

1) Assuming that for small enough balls we can ensure that the topological boundary of our curve is outside such ball (as required by the *monotonicity lemma*) how do we deal with the fact that the target manifold is not compact? That's the key assumption in the lemma. As I wrote in the comments I though that we could just restrict our attention to some compact sub manifolds of $\Sigma' \cup \partial \Sigma' \times [0,\infty)$ but the lower bound provided by the lemma would depend on such a choice. Hence it's not clear to me how to obtain a uniform lower bound on the energy of the curve passing through a given ball.

2)If we work with a vertical almost complex structure on the neck, then I think the lower bound provided by the monotonicity lemma is the same in every sector $[a-n,a+n]\times \partial \Sigma'$ (since they all are isometric and with the same a.c.s.) In that case I can see how to get a uniform lower bound. But what worries me is that I don't see any problem in applying this reasoning to case #1, but as far as I understood, in that case we can't rule out non-constant $j$-hol planes that easily and we must use a different strategy (namely integrability of the a.c.s.) 



  [1]: https://pdfs.semanticscholar.org/64f9/126064e02574bbc9de79608304264b09fbfb.pdf