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In the papers https://arxiv.org/abs/math/0410059, https://arxiv.org/abs/math/0410061, https://arxiv.org/abs/1608.07988; Hutchings, Sullivan and Choi use a interesting trick to dismiss the existence of pseudoholomorphic curves counted by the ECH differential. That is, they are in a Morse-Bott degenerate setting, so they perturb the contact form to get only two "non-degenerate" Reeb orbits (an elliptic and a positive hyperbolic orbits) for every foliated tori, and they do this for all tori with "energy" less that a fixed $L$, then make this $L\rightarrow \infty$ and you get a sequence of contact forms converging to the initial one (See page 9 of https://arxiv.org/pdf/1608.07988.pdf; for example).

Anyway, to do this perturbation in every such tori, Morse functions in $S^1$ are required, they are carefully chosen so that the (only two) extremal values of the function be at "specific" places; this "specificity" is made clear by parametrizing the space of Reeb orbits in any such foliated tori (see (57) in Lemma A.1 of https://arxiv.org/abs/math/0410059 and the proof therein for example).

It happens that in such a perturbative setting some necessary conditions can be found for the pseudoholomorphic curves counted by the ECH differential to meet, however some of the curves satisfying these conditions can actually be eliminated of the count by perturbing in "the right way" and by this I mean using the specific choices of the Morse functions.

Now, here are my questions:

  1. This seems to work only when our contact 3-manifold is the product of three 1-manifolds (or at least is like that up to a pair of Reeb orbits, because I know $S^3$ and the lens spaces, can be treated in the same way, see https://arxiv.org/abs/1608.07988; for example). Can this be recreated for more general cases, for example a case where the 3-manifold is "piece-wise" $I\times T^2$ (for example, three of these pieces) and these pieces meet at critical levels where there are other Reeb orbits, not contained on foliated tori?

  2. Is there a more general technique for dismissing existence of Pseudoholomorphic curves? I have seen the papers https://arxiv.org/abs/math/0701300, https://arxiv.org/abs/0705.2074 on obstruction bundle gluing, which in theory, could always be used for dismissing or proving existence of pseudoholomorphic curves in the symplectization of 3-manifolds setting. The way I see it, proving that $G(u_1,u_2)=0$ (see https://arxiv.org/abs/math/0701300 page 9 for the notation) for a pseudoholomorphic building with levels formed by $u_1$ and $u_2$ should imply the non-existence of pseudoholomorphic curves (with certain properties in relation to the ends of the building) near the building. So, could this be effectively used for a specific problem? For example, could we redo the result in the first three papers with this idea instead of using the parametrization of the Reeb orbits in a foliated tori? I asked this because there are cases in which this parametrization cannot be attained in global coordinates nor will cover all Reeb orbits, for example, the orbits outside the $I\times T^2$ pieces. Notice that indeed, the gluing papers came like four years after the papers on PFH and ECH of $T^3$, so the idea may not be far-fetched.

I appreciate any help, thanks in advance.

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  • $\begingroup$ Have you tried asking Michael Hutchings himself? $\endgroup$ Nov 25 at 17:48
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    $\begingroup$ I tried to reach him but haven't got any answer, I thought maybe could find another expert here... $\endgroup$
    – kvicente
    Nov 25 at 22:10
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(1) It cannot necessarily be done in general as you suggest, because you don’t have control over the “other Reeb orbits” so there are a priori bad curves that can hit them. It does however work on some contact 3-manifolds, such as S1xS2 with a certain overtwisted contact form (see my papers on SW = Gr)… in fact, this example goes back to Taubes which is referenced by Hutchings-Sullivan in their T3 paper.

As for (2), it’s mostly a game of combinatorics and systems of equations. There are a bunch of index inequalities that must hold (involving the standard Fredholm index and the magical ECH index), and curves counted by the ECH differential (or U-maps and loop-maps) must satisfy certain “partition conditions” of its asymptotics.

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  • $\begingroup$ Thanks @Chris_Gerig, do you know any paper where OBG is used to study the ECH of an specific example? Has somebody done that? $\endgroup$
    – kvicente
    Nov 30 at 12:01

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