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By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The monotonicity lemma now prevents the sequence of rescaled holomorphic discs from "thinning" out and having noncompact image. That is, our maps must use up at least a certain amount of energy for every ball whose center it passes through. The version of the monotonicity lemma used here is not just for compact manifolds such as $(\Sigma',\omega)$ but also symplectizations such as $(\partial\Sigma'\times[0,\infty),d\theta\wedge ds)$, see Proposition 2.71 of Abbas' book "An introduction to compactness results in symplectic field theory". We couldn't a priori apply the usual monotonicity lemma to each compact set in a compact exhaustion of the noncompact surface, because the resulting (lower) energy bound may vary (hence decrease) with each set.

Restating: The boundedness assumption means precisely that our bubble points are stuck in $\Sigma'$ union a collar $\partial\Sigma'\times[0,k]$ and cannot escape along $\partial\Sigma'\times[0,\infty)$, otherwise we're in Case#1 (in the paper) which has the points running away along the neck of $\Sigma^{R_i}$. As for the rest of the images of our maps, given balls centered around various points in our maps' image, monotonicity prevents the maps from having arbitrarily small area in any of these balls and hence the images of the maps are contained in some overall compact region.

  Note: In both the case at hand (Case#2 in the paper) and Case#1 in the paper, our sequence of rescaled maps have domains being open disks centered around the bubble points. It can happen in Case#1 that these disks just shift along with the bubble points down the infinite neck region, which doesn't happen in Case#2.

It turns out that the Removable Singularities Theorem and the Monotonicity Lemma do not require compactness, but that the target manifold should have bounded geometry, such as in our case of inserting a cylindrical piece to the compact surface. See for example, Lemma 5.11 in "Compactness results in symplectic field theory" by Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder and Proposition 2.71 of Abbas' book "An introduction to compactness results in symplectic field theory". Seidel brought to my attention Theorem 4.5.1 in Sikorav's paper "Some properties of holomorphic curves in almost complex manifolds", which only needs the (possibly noncompact) target to have bounded geometry and does not need the disc to have bounded image, for removing singularities.

By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$, i.e. our bubble points are stuck in $\Sigma'$ union a collar $\partial\Sigma'\times[0,k]$ and cannot escape along $\partial\Sigma'\times[0,\infty)$, otherwise we're in Case#1 (in the paper) which has the points running away along the neck of $\Sigma^{R_i}$. We now zoom in our around these bubble points (hence away from the boundary of the holomorphic strips), forming a rescaled sequence of holomorphic discs. Note that in both the case at hand (Case#2 in the paper) and Case#1 in the paper, our sequence of rescaled maps have domains being open disks centered around the bubble points. It can happen in Case#1 that these disks just shift along with the bubble points down the infinite neck region, which doesn't happen in Case#2.

Aside: I hoped that the monotonicity lemma would prevent the sequence of rescaled holomorphic discs from "thinning" out and having unbounded image, that is, our maps must use up at least a certain amount of energy for every ball whose center it passes through. We couldn't a priori apply the usual monotonicity lemma (for compact target manifolds) to each compact set in a compact exhaustion of the noncompact surface, because the resulting (lower) energy bound may vary (hence decrease) with each set. But due to the cylindrical end there is a uniform lower bound. In any case, the a priori issue is that the (image of the) boundary of the discs might squeeze into the balls with which we are trying to compute the area bounds.

By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The monotonicity lemma now prevents the sequence of rescaled holomorphic discs from "thinning" out and having noncompact image. That is, our maps must use up at least a certain amount of energy for every ball whose center it passes through. The version of the monotonicity lemma used here is not just for compact manifolds such as $(\Sigma',\omega)$ but also symplectizations such as $(\partial\Sigma'\times[0,\infty),d\theta\wedge ds)$, see Proposition 2.71 of Abbas' book "An introduction to compactness results in symplectic field theory".

Restating: The boundedness assumption means precisely that our bubble points are stuck in $\Sigma'$ union a collar $\partial\Sigma'\times[0,k]$ and cannot escape along $\partial\Sigma'\times[0,\infty)$, otherwise you're in Case#1 (in the paper) which has the points running away along the neck of $\Sigma^{R_i}$. As for the rest of the images of the maps, given balls centered around various points in your maps' image, monotonicity prevents the maps from having arbitrarily small area in any of these balls and hence the images of the maps are contained in some overall compact region.

Note: In both the case at hand (Case#2 in the paper) and Case#1 in the paper, our sequence of rescaled maps have domains being open disks centered around the bubble points. It can happen in Case#1 that these disks just shift along with the bubble points down the infinite neck region, which doesn't happen in Case#2.

By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The monotonicity lemma now prevents the sequence of rescaled holomorphic discs from "thinning" out and having noncompact image. That is, our maps must use up at least a certain amount of energy for every ball whose center it passes through. The version of the monotonicity lemma used here is not just for compact manifolds such as $(\Sigma',\omega)$ but also symplectizations such as $(\partial\Sigma'\times[0,\infty),d\theta\wedge ds)$, see Proposition 2.71 of Abbas' book "An introduction to compactness results in symplectic field theory". We couldn't a priori apply the usual monotonicity lemma to each compact set in a compact exhaustion of the noncompact surface, because the resulting (lower) energy bound may vary (hence decrease) with each set.

Restating: The boundedness assumption means precisely that our bubble points are stuck in $\Sigma'$ union a collar $\partial\Sigma'\times[0,k]$ and cannot escape along $\partial\Sigma'\times[0,\infty)$, otherwise we're in Case#1 (in the paper) which has the points running away along the neck of $\Sigma^{R_i}$. As for the rest of the images of our maps, given balls centered around various points in our maps' image, monotonicity prevents the maps from having arbitrarily small area in any of these balls and hence the images of the maps are contained in some overall compact region.

Note: In both the case at hand (Case#2 in the paper) and Case#1 in the paper, our sequence of rescaled maps have domains being open disks centered around the bubble points. It can happen in Case#1 that these disks just shift along with the bubble points down the infinite neck region, which doesn't happen in Case#2.

By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The monotonicity lemma now prevents the sequence of renormalized holomorphic discs from "thinning" out and having noncompact image. That is, our maps must use up at least a certain amount of energy for every ball whose center it passes through.

Restating: The boundedness assumption means precisely that our bubble points are stuck in $\Sigma'$ union a collar $\partial\Sigma'\times[0,k]$ and cannot escape along $\partial\Sigma'\times[0,\infty)$, otherwise you're in Case#1 (in the paper) which has the points running away along the neck of $\Sigma^{R_i}$. As for the rest of the images of the maps, given balls centered around various points in your maps' image, monotonicity prevents the maps from having arbitrarily small area in any of these balls and hence the images of the maps are contained in some overall compact region.

By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The monotonicity lemma now prevents the sequence of rescaled holomorphic discs from "thinning" out and having noncompact image. That is, our maps must use up at least a certain amount of energy for every ball whose center it passes through. The version of the monotonicity lemma used here is not just for compact manifolds such as $(\Sigma',\omega)$ but also symplectizations such as $(\partial\Sigma'\times[0,\infty),d\theta\wedge ds)$, see Proposition 2.71 of Abbas' book "An introduction to compactness results in symplectic field theory".

Restating: The boundedness assumption means precisely that our bubble points are stuck in $\Sigma'$ union a collar $\partial\Sigma'\times[0,k]$ and cannot escape along $\partial\Sigma'\times[0,\infty)$, otherwise you're in Case#1 (in the paper) which has the points running away along the neck of $\Sigma^{R_i}$. As for the rest of the images of the maps, given balls centered around various points in your maps' image, monotonicity prevents the maps from having arbitrarily small area in any of these balls and hence the images of the maps are contained in some overall compact region.

Note: In both the case at hand (Case#2 in the paper) and Case#1 in the paper, our sequence of rescaled maps have domains being open disks centered around the bubble points. It can happen in Case#1 that these disks just shift along with the bubble points down the infinite neck region, which doesn't happen in Case#2.