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In the papers The periodic Floer homology of a Dehn twist, Rounding corners of polygons and the embedded contact homology of $T^3$, and Combinatorial embedded contact homology for toric contact manifolds, Hutchings and Sullivan, and Choi, use an 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 Choi - Combinatorial embedded contact homology for toric contact manifolds for example).

Anyway, to do this perturbation in every such torus, Morse functions in $S^1$ are required, they are carefully chosen so that the (only two) extremal values of the function are 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 Hutchings and Sullivan - The periodic Floer homology of a Dehn twist 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 Choi - Combinatorial embedded contact homology for toric contact manifolds 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 Hutchings and Taubes - Gluing pseudoholomorphic curves along branched covered cylinders I, Hutchings and Taubes - Gluing pseudoholomorphic curves along branched covered cylinders II 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 Hutchings and Taubes - Gluing pseudoholomorphic curves along branched covered cylinders I 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.

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  • $\begingroup$ Have you tried asking Michael Hutchings himself? $\endgroup$ Commented Nov 25, 2021 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
    Commented Nov 25, 2021 at 22:10

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  1. Just some context of "what it means to choose generic Morse–Bott perturbation": So when doing Morse–Bott theory what you are secretly doing in the Morse–Bott picture is counting cascades (objects that are called "buildings" on page 10 of Choi - Combinatorial embedded contact homology for toric contact manifolds. This is not the same thing as an SFT building!) that are cut out transversely (they live in a some kind of "fiber product" of moduli spaces, and for those spaces there is a notion of transversality/Fredholm index, given suitable assumptions). Transversality is achieved by choosing generic $J$. So after fixing a Morse–Bott perturbation of contact form, $J$ holomorphic curves that "accidentally meet in a way that makes them a cascade" but don't live in a transversely cut out "fiber product" should disappear after choosing generic $J$ (and therefore should not be counted). This is similar to not counting curves that can be eliminated by choosing the Morse–Bott perturbation generically enough (I think in the papers you cite only for very special $J$ can we enumerate all the $J$ curves, so they want to avoid changing this $J$ so they move around the Morse–Bott perturbation instead).

  2. As for cascades with levels that are not Morse–Bott tori but nondegenerate Reeb orbits, I think (but don't yet have proof) this might make the index too high (ECH index/Fredholm index) for example a building consisting of curves $u_1$, $u_2$ that meet along a single nondegenerate Reeb orbit has ECH index and Fredholm index the sum of that of $u_1$ and $u_2$.

  3. The 2 obstruction bundle gluing papers do indeed work in an if and only if fashion, but be careful with what they mean by $G=0$, because there is a signed count of holomorphic curves. When they deform the nonlinear obstruction section to the linear obstruction section only the signed count of holomorphic curves stays the same, so it could be the case there are 200 holomorphic curves but they all cancel in pairs. To use these papers to rule out all curves you need to show the nonlinear obstruction section is never zero, which is a much more daunting task.

I have not yet seen any people ruling out curves using obstruction bundle gluing (in symplectizations at least, in the compact case I think it's different). From my limited experience it's usually much easier to first try ruling out curves from intersection theory/energy considerations than using analysis directly.

Edit: also I don't think the results in obstruction bundle papers can be used in the earlier papers you mentioned with the specific lemmas mentioned as end result. As I mentioned in bullet 1 the technique for choosing generic Morse–Bott perturbation is more like choosing generic $J$, but obstruction bundle gluing papers deal with something else entirely.

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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
    Commented Nov 30, 2021 at 12:01

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