Timeline for Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends
Current License: CC BY-SA 4.0
18 events
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| Jan 29, 2020 at 14:52 | vote | accept | Riccardo | ||
| Dec 4, 2019 at 16:11 | history | edited | Chris Gerig | CC BY-SA 4.0 |
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| Dec 3, 2019 at 3:35 | comment | added | Riccardo | Ok, let me rephrase. Integrability implies that that the limit cannot be a non constant J-hol sphere or plane. I was wondering if we could rule out at least J-hol planes in case #1 without appealing to integrability by using monotonicity. Sphere can be rule out by means of exactness of the symplectic form on the neck . | |
| Dec 3, 2019 at 2:51 | comment | added | Riccardo | There are not, by the maximum principle (since they would factor through the universal cover of the cylinder) I'd say. I was just wondering if the argument I wrote was still correct though, since it seems "more general" (not relying on the very peculiar case of working with an integrable complex structure that is something typical in dimension 2) | |
| Dec 3, 2019 at 1:15 | comment | added | Riccardo | @user_1789 I was wondering if we could just take a bigger disk. As far as I understood, the proof should be the same, I just get a bigger constant. The only thing that is unclear to me is that, while it's clear that we can assume that $\text{Im}(\tilde{u}(D))\not\subset B_r$, I don't know if we can assume that for big enough disk all the boundary is going to be outside it. Seems like the case but I can't find a precise way to show it | |
| Dec 3, 2019 at 0:18 | comment | added | user_1789 | @ChrisGerig How do you ensure that $\tilde{u}(I_1)\cap B_r(\tilde{u}(z_0))=\emptyset$ in the notation of Abbas? | |
| Dec 3, 2019 at 0:17 | comment | added | Riccardo | hence my curve is contained in the region $S^1\times [-2Kr,2Kr]\times S^1$ (translated by $\tilde{u}(0)$) which is compact. | |
| Dec 3, 2019 at 0:17 | comment | added | Riccardo | yes your clarifications were very helpful. I think I concentrate all my remaining doubts in the following: what prevents from using monotonicity (the very same argument) in case #1? I would proceed as follows. let $\tilde{u}$ my limiting curve from $\Bbb C$ into $S^1\times (-\infty, \infty)$, it has finite energy, i take any path $\gamma$ from $0$ to $\pm \infty$ in the domain a collection of balls of radius $r$ whose centre lies on $u(\gamma)$ and they don't intersect (like a string of pearls). by the monotonicity I know I can have at most $K<\infty$ of them, | |
| Dec 2, 2019 at 21:49 | history | edited | Chris Gerig | CC BY-SA 4.0 |
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| Dec 2, 2019 at 21:29 | comment | added | Riccardo | Thanks for pointing out this version of the monotonicity lemma, I was unaware of it. I'm kind of worried about it's assumptions though. how do we ensure that the sectors in the boundary of the disk are mapped over legendrian sub-manifolds in the neck? | |
| Dec 2, 2019 at 20:18 | history | edited | Chris Gerig | CC BY-SA 4.0 |
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| Nov 27, 2019 at 5:04 | vote | accept | Riccardo | ||
| Dec 2, 2019 at 14:37 | |||||
| Nov 27, 2019 at 4:32 | history | edited | Chris Gerig | CC BY-SA 4.0 |
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| Nov 27, 2019 at 3:57 | history | edited | Chris Gerig | CC BY-SA 4.0 |
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| Nov 27, 2019 at 2:50 | history | edited | Chris Gerig | CC BY-SA 4.0 |
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| Nov 27, 2019 at 2:48 | history | undeleted | Chris Gerig | ||
| Nov 27, 2019 at 2:40 | history | deleted | Chris Gerig | via Vote | |
| Nov 27, 2019 at 2:36 | history | answered | Chris Gerig | CC BY-SA 4.0 |