This question is related to my old question Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends about the bubbling off argument in Seidel's paper The symplectic Floer homology of a Dehn twist in the case of a surface.

I was trying to find a reason for the following seemingly obvious passage:

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 $\Sigma'=\Sigma \setminus [-1,1]\times S^1$

At this point we have two cases:

1. The distance of $u^i(z_i)$ from $\Sigma'\subset \Sigma^{R_i}$ goes to infinity.
2. Such distance is bounded.

My previous question was about case 2, now I'm actually curious about the fine details of case 1. In that case the author concludes that we would have a finite energy "bubble" entirely contained in the cylinder $\Bbb R \times S^1$ and then he proceed in ruling that out.

My question now is the following: how can we ensure that in case 1, the bubble is entirely contained in the cylinder?

I guess using some SFT reasoning we can analysing the pseudo-holomorphic building that the sequence $\{u^i\}$ is limiting to and conclude that finite energy components cannot go through different "stories" (since it would cost too much energy). But I'm sure the authors had in mind a much simpler reasoning for that, since the compactness SFT papers came out several years after this computation about the Floer homology of a Dehn twist. The usual rescaling argument implies that the image of the rescaled maps $$ u^i(C_i^{-1}z+z_i) : D_{C_i}(0)\to \Sigma^{R_i}$$ coincides with the image of the restriction of the maps to the unit disk $$B_i:=u^i(D_1(z_i))$$ and therefore the image of my limit bubble $\tilde{u}$ will be the limit of these $B_i$ in the split manifold $\Sigma'\cup \Bbb R_{\pm}\times S^1 \cup \Bbb R \times S^1$.

I honestly don't see how to show that the limit of these $B_i$ will be entirely contained in $\Bbb R \times S^1$, even though it looks like a consequence of the uniform bound on their area. If we could prevent that these "blobs" stretch out and exit the collar $[-R_i,R_i]\times S^1$ we would be in business, but usual monotonicity arguments can't work here because for each step $i$, the relevant constants might change (see the discussion in the linked question for example).

Is there an easy observation that proves that the sphere must be entirely contained in the neck?

This question is related to my old question Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends about the bubbling off argument in Seidel's paper The symplectic Floer homology of a Dehn twist in the case of a surface.

I was trying to find a reason for the following seemingly obvious passage:

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 $\Sigma'=\Sigma \setminus [-1,1]\times S^1$

At this point we have two cases:

1. The distance of $u^i(z_i)$ from $\Sigma'\subset \Sigma^{R_i}$ goes to infinity.
2. Such distance is bounded.

My previous question was about case 2, now I'm actually curious about the fine details of case 1. In that case the author concludes that we would have a finite energy "bubble" entirely contained in the cylinder $\Bbb R \times S^1$ and then he proceed in ruling that out.

My question now is the following: how can we ensure that in case 1, the bubble is entirely contained in the cylinder?

The usual rescaling argument implies that the image of the rescaled maps $$ u^i(C_i^{-1}z+z_i) : D_{C_i}(0)\to \Sigma^{R_i}$$ coincides with the image of the restriction of the maps to the unit disk $$B_i:=u^i(D_1(z_i))$$ and therefore the image of my limit bubble $\tilde{u}$ will be the limit of these $B_i$ in the split manifold $\Sigma'\cup \Bbb R_{\pm}\times S^1 \cup \Bbb R \times S^1$.

I honestly don't see how to show that the limit of these $B_i$ will be entirely contained in $\Bbb R \times S^1$.

This is what I tried: let $$ d_i := d(u^i(z_i)), \Sigma')\to \infty$$ If $$\lim_{i\to \infty} \frac{d_i}{C_i}>0$$ then the usual rescaling/bubbling reasoning works fine and gives a bubble completely contained in the neck. If $$\lim_{i\to \infty} \frac{d_i}{C_i}=0$$ I run into some troubles in showing that the usual rescaling works. In fact, I have to restrict the functions $u^i$ to a domain contained in the disks $D_{\frac{d_i}{C_i}}(z_i)$ to ensure that the maps land in the neck. But those disks are shrinking! can I still apply the usual bubbling argument even if the domain of my family of functions is degenerating to a point? I would have the following rescaled/restricted functions $$ \tilde{u}^i(\frac{z}{C_i}+ z_i) \colon D_{d_i}(0)\to [-R_i,R_i]\times S^1$$ and they have uniformly bounded $L^{\infty}$-norm and their domain is approaching $\Bbb C$. So it seems to me we have a "bubble". My only problem is that all the references start by working with functions with fixed domain, and not one that is degenerating to a point

this question is related to my old question about the bubbling off argument in Seidel's paper about the Floer cohomology of a Dehn twist in the case of a surface.

I was trying to find a reason for the following seemingly obvious passage:

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 $\Sigma’=\Sigma \setminus [-1,1]\times S^1$

At this point we have two cases:

1. case 1. the distance of $u^i(z_i)$ from $\Sigma'\subset \Sigma^{R_i}$ goes to infinity
2. case 2. such distance is bounded.

My previous question was about case 2, now I'm actually curious about the fine details of case 1. In that case the author concludes tat we would have a finite energy "bubble" entirely contained in the cylinder $\Bbb R \times S^1$ and then he proceed in ruling that out.

My question now is the following: how can we ensure that in case 1, the bubble is entirely contained in the cylinder?

I guess using some SFT reasoning we can analysing the pseudo-holomorphic building that the sequence $\{u^i\}$ is limiting to and conclude that finite energy components cannot go through different "stories" (since it would cost too much energy). but I'm sure the authors had in mind a much simpler reasoning for that, since the compactness SFT papers came out several years after this computation about the Floer homology of a Dehn twist. The usual rescaling argument implies that the image of the rescaled maps $$ u^i(C_i^{-1}z+z_i) : D_{C_i}(0)\to \Sigma^{R_i}$$ coincides with the image of the restriction of the maps to the unit disk $$B_i:=u^i(D_1(z_i))$$ and therefore the image of my limit bubble $\tilde{u}$ will be the limit of these $B_i$ in the split manifold $\Sigma'\cup \Bbb R_{\pm}\times S^1 \cup \Bbb R \times S^1$.

I honestly don't see how to show that the limit of these $B_i$ will be entirely contained in $\Bbb R \times S^1$, even though it looks like a consequence of the uniform bound on their area. If we could prevent that these "blobs" stretch out and exit the collar $[-R_i,R_i]\times S^1$ we would be in business, but usual monotonicity arguments can't work here because for each step $i$, the relevant constants might change (see the discussion in the liked question for example).

Is there an easy observation that proves that the sphere must be entirely contained in the neck?

This question is related to my old question Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends about the bubbling off argument in Seidel's paper The symplectic Floer homology of a Dehn twist in the case of a surface.

I was trying to find a reason for the following seemingly obvious passage:

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 $\Sigma'=\Sigma \setminus [-1,1]\times S^1$

At this point we have two cases:

1. The distance of $u^i(z_i)$ from $\Sigma'\subset \Sigma^{R_i}$ goes to infinity.
2. Such distance is bounded.

My previous question was about case 2, now I'm actually curious about the fine details of case 1. In that case the author concludes that we would have a finite energy "bubble" entirely contained in the cylinder $\Bbb R \times S^1$ and then he proceed in ruling that out.

My question now is the following: how can we ensure that in case 1, the bubble is entirely contained in the cylinder?

I guess using some SFT reasoning we can analysing the pseudo-holomorphic building that the sequence $\{u^i\}$ is limiting to and conclude that finite energy components cannot go through different "stories" (since it would cost too much energy). But I'm sure the authors had in mind a much simpler reasoning for that, since the compactness SFT papers came out several years after this computation about the Floer homology of a Dehn twist. The usual rescaling argument implies that the image of the rescaled maps $$ u^i(C_i^{-1}z+z_i) : D_{C_i}(0)\to \Sigma^{R_i}$$ coincides with the image of the restriction of the maps to the unit disk $$B_i:=u^i(D_1(z_i))$$ and therefore the image of my limit bubble $\tilde{u}$ will be the limit of these $B_i$ in the split manifold $\Sigma'\cup \Bbb R_{\pm}\times S^1 \cup \Bbb R \times S^1$.

I honestly don't see how to show that the limit of these $B_i$ will be entirely contained in $\Bbb R \times S^1$, even though it looks like a consequence of the uniform bound on their area. If we could prevent that these "blobs" stretch out and exit the collar $[-R_i,R_i]\times S^1$ we would be in business, but usual monotonicity arguments can't work here because for each step $i$, the relevant constants might change (see the discussion in the linked question for example).

Is there an easy observation that proves that the sphere must be entirely contained in the neck?

this question is related to my old question about the bubbling off argument in Seidel's paper about the Floer cohomology of a Dehn twist in the case of a surface.

I was trying to find a reason for the following seemingly obvious passage:

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 $\Sigma’=\Sigma \setminus [-1,1]\times S^1$

At this point we have two cases:

1. case 1. the distance of $u^i(z_i)$ from $\Sigma'\subset \Sigma^{R_i}$ goes to infinity
2. case 2. such distance is bounded.

My previous question was about case 2, now I'm actually curious about the fine details of case 1. In that case the author concludes tat we would have a finite energy "bubble" entirely contained in the cylinder $\Bbb R \times S^1$ and then he proceed in ruling that out.

My question now is the following: how can we ensure that in case 1, the bubble is entirely contained in the cylinder?

I guess using some SFT reasoning we can analysing the pseudo-holomorphic building that the sequence $\{u^i\}$ is limiting to and conclude that finite energy components cannot go through different "stories" (since it would cost too much energy). but I'm sure the authors had in mind a much simpler reasoning for that, since the compactness SFT papers came out several years after this computation about the Floer homology of a Dehn twist. The usual rescaling argument implies that the image of the rescaled maps $$ u^i(C_i^{-1}z+z_i) : D_{C_i}(0)\to \Sigma^{R_i}$$ coincides with the image of the restriction of $B_i:=u^i(D_1(z_i))$, and therefore the image of my limit bubble $\tilde{u}$ will be the limit of these $B_i$ in the split manifold $\Sigma'\cup \Bbb R_{\pm}\times S^1 \cup \Bbb R \times S^1$.

I honestly don't see how to show that the limit of these $B_i$ will be entirely contained in $\Bbb R \times S^1$, even though it looks like a consequence of the uniform bound on their area. If we could prevent that these "blobs" stretch out and exit the collar $[-R_i,R_i]\times S^1$ we would be in business, but usual monotonicity arguments can't work here because for each step $i$, the relevant constants might change (see the discussion in the liked question for example).

Is there an easy observation that proves that the sphere must be entirely contained in the neck?

this question is related to my old question about the bubbling off argument in Seidel's paper about the Floer cohomology of a Dehn twist in the case of a surface.

I was trying to find a reason for the following seemingly obvious passage:

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 $\Sigma’=\Sigma \setminus [-1,1]\times S^1$

At this point we have two cases:

1. case 1. the distance of $u^i(z_i)$ from $\Sigma'\subset \Sigma^{R_i}$ goes to infinity
2. case 2. such distance is bounded.

My previous question was about case 2, now I'm actually curious about the fine details of case 1. In that case the author concludes tat we would have a finite energy "bubble" entirely contained in the cylinder $\Bbb R \times S^1$ and then he proceed in ruling that out.

My question now is the following: how can we ensure that in case 1, the bubble is entirely contained in the cylinder?

I guess using some SFT reasoning we can analysing the pseudo-holomorphic building that the sequence $\{u^i\}$ is limiting to and conclude that finite energy components cannot go through different "stories" (since it would cost too much energy). but I'm sure the authors had in mind a much simpler reasoning for that, since the compactness SFT papers came out several years after this computation about the Floer homology of a Dehn twist. The usual rescaling argument implies that the image of the rescaled maps $$ u^i(C_i^{-1}z+z_i) : D_{C_i}(0)\to \Sigma^{R_i}$$ coincides with the image of the restriction of the maps to the unit disk $$B_i:=u^i(D_1(z_i))$$ and therefore the image of my limit bubble $\tilde{u}$ will be the limit of these $B_i$ in the split manifold $\Sigma'\cup \Bbb R_{\pm}\times S^1 \cup \Bbb R \times S^1$.

I honestly don't see how to show that the limit of these $B_i$ will be entirely contained in $\Bbb R \times S^1$, even though it looks like a consequence of the uniform bound on their area. If we could prevent that these "blobs" stretch out and exit the collar $[-R_i,R_i]\times S^1$ we would be in business, but usual monotonicity arguments can't work here because for each step $i$, the relevant constants might change (see the discussion in the liked question for example).

Is there an easy observation that proves that the sphere must be entirely contained in the neck?