Timeline for Varying a $J$-holomorphic sphere in a symplectic $4$-manifolds
Current License: CC BY-SA 3.0
15 events
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| Nov 2, 2017 at 22:00 | comment | added | Nikolaki | Check out the article link.springer.com/article/10.1007/BF02921708 by Hofer-Lizan-Sikorav on automatic transversality, and in particular the proposition on the bottom of p. 155. This tells you that the infinitessimal variations are of the correct dimension, and also non-vanishing, thus giving rise to the sought smooth foliation. | |
| S Oct 28, 2017 at 17:21 | history | bounty ended | CommunityBot | ||
| S Oct 28, 2017 at 17:21 | history | notice removed | CommunityBot | ||
| Oct 20, 2017 at 15:24 | history | edited | aglearner | CC BY-SA 3.0 |
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| S Oct 20, 2017 at 15:24 | history | bounty started | aglearner | ||
| S Oct 20, 2017 at 15:24 | history | notice added | aglearner | Draw attention | |
| Oct 19, 2017 at 11:25 | comment | added | Chris Gerig | When you vary $J$ the Fredholm index of the deformation operator of the curve doesn't change (functional analysis), so transversality still holds (this is "generic transversality" in McDuff-Salamon's book on $J$-holomorphic curves). | |
| Oct 19, 2017 at 10:56 | comment | added | aglearner | Chris thanks, I agree this should be correct, it is just that I need to use this statement in a paper and want to have a proper reference (which I understand a bit). Sikorav and Co do not vary J... I imagine this should be fairly straight-forward for those who practice $J$-holomorphic curves... | |
| Oct 19, 2017 at 10:03 | comment | added | Chris Gerig | I don't know if it does follow, but it's what popped into my mind (a reference is given in another MO question of yours: J on $CP^2$ that are not tamed). It says that when I perturb $J$ to $J'$ there is a $J'$-holomorphic sphere isotopic to $S^2_0$ (call it still $S^2_0$). So we get the 1-parameter family of $J$-holomorphic spheres in $M\times(-\epsilon,\epsilon)$. I think if $S^2_0$ has a neighborhood $U_0$ in $M$ for which there are unique $J$-holomorphic spheres through each point of the neighborhood, we would get the same statement in the neighborhood $U_0\times(-\epsilon,\epsilon)$. | |
| Oct 19, 2017 at 9:57 | comment | added | aglearner | Dear Chris, many thanks for your comment! Do you think you can give me some relatively pedagogical (or relatively precise) reference from which it will be clear that the statement indeed follows from "automtatic transversality". Or maybe develop your comment into a slightly longer answer? My knowledge in PDEs is poor... I guess I would like to have something line "automatic transverality with parameters, because usually it is stated for $J$ fixed | |
| Oct 19, 2017 at 8:11 | history | edited | aglearner | CC BY-SA 3.0 |
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| Oct 19, 2017 at 1:40 | comment | added | Chris Gerig | Why wouldn't this follow immediately from "automatic transversality"? | |
| Oct 19, 2017 at 0:27 | history | edited | aglearner | CC BY-SA 3.0 |
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| Oct 18, 2017 at 14:02 | history | edited | aglearner | CC BY-SA 3.0 |
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| Oct 18, 2017 at 13:55 | history | asked | aglearner | CC BY-SA 3.0 |