Hidden branched covers of trivial cylinders

I would now like to explain another ECH-type trick, which I have been meaning to write about here for a while, and which may have applications to other kinds of contact homology in three dimensions and holomorphic curve counts in four dimensions.

1. The general situation.

Suppose we are trying to prove some kind of compactness for some kind of holomorphic curves in the symplectization of a contact three-manifold, or a symplectic cobordism between contact three-manifolds. We know from the general compactness result of Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder that any sequence of holomorphic curves (of bounded symplectic area and genus between the same sets of Reeb orbits) has a subsequence which converges to a “holomorphic building”. A component of this building in a symplectization level may include a branched cover of a trivial cylinder. (Here, a “trivial cylinder” in the symplectization of a contact manifold $Y$ is a cylinder of the form ${\mathbb R}\times \gamma\subset{\mathbb R}\times Y$ where $\gamma$ is an embedded Reeb orbit in $Y$.)

For example, in the proof that $\partial^2=0$ in embedded contact homology, the limit of a sequence of holomorphic curves with ECH index $2$ may include branched covers of trivial cylinders in between curves with ECH index $1$. This is why the pair of papers with Taubes that prove that $\partial^2=0$ in ECH (as well as a more general gluing theorem) is 200 pages, instead of just a paragraph quoting previous gluing theorems.

Now a somewhat worse situation is when a branched cover of a trivial cylinder appears at the top or bottom of the holomorphic building. However I claim that this more or less never happens! In other words, branched covers of trivial cylinders stay “hidden” between other levels of the limiting building, and are never “exposed” at the top or bottom of the building.

I don’t have a general proof (or precise statement of the hypotheses) of this claim, but I can prove it in some cases, sometimes modulo some analysis which still needs to be worked out. I will now explain how to completely prove this claim in a very special case, which however is of some interest.

2. The special case.

Let $Y=S^3$, and let $\lambda$ be a nondegenerate, dynamically convex contact form on $S^3$. Recall that the term “dynamically convex”, going back to Hofer-Wysocki-Zehnder, means that each Reeb orbit $\gamma$ has Conley-Zehnder index $CZ(\gamma)\ge 3$. Here we define the Conley-Zehnder index of $\gamma$ using a trivialization of the restriction of the contact plane field to $\gamma$ that extends over a disk bounded by $\gamma$.

In this situation, one would like to define the cylindrical contact homology of $(S^3,\lambda)$, using dynamical convexity to rule out bubbling of holomorphic planes. That is, in the compactness arguments to prove that the cylindrical contact homology differential $\partial$ is well-defined and satisfies $\partial^2=0$, one has to worry about convergence to holomorphic buildings including holomorphic planes, together with some other holomorphic curves that are genus zero and have one positive but arbitrarily many negative ends. Dynamical convexity implies that every holomorphic plane has Fredholm index at least $2$.  If the other curves in the building are cut out transversely, then the total Fredholm index of the building will be too big, so this kind of degeneration will be ruled out.

3. The problem.

It may be possible to arrange transversality for the relevant holomorphic curves that are not branched covers of trivial cylinders. The paper by Bourgeois-Cieliebak-Ekholm arranges this transversality using $S^1$-dependent almost complex structures, but I don’t see how they get around the other problems with $S^1$-dependent almost complex structures that I have described in recent posts. In fact, there is some hope that the relevant transversality in the symplectization may work for a generic $S^1$-independent almost complex structure. Let’s suppose for the sake of argument that this works, or that we have arranged the necessary transversality some other way. We then get a well-defined differential $\partial$.

There is now a second problem, involving branched covers of trivial cylinders, that arises when one tries to prove that $\partial^2=0$. Specifically, a sequence of index $2$ cylinders may converge to a holomorphic building of the following type. There are two levels $u_1$ and $u_2$. The upper level $u_1$ is an index zero pair of pants with one positive end and two negative ends which is a branched cover of degree $m+1$ of a trivial cylinder ${\mathbb R}\times\gamma$, where $\gamma$ is an embedded elliptic Reeb orbit with monodromy angle $\theta\in(1,2)$ (with respect to the usual trivialization), and $m$ is a positive integer. The lower level $u_2$ has two components. One component is a cylinder which is a degree $m$ cover of ${\mathbb R}\times\gamma$. The other component is a somewhere injective index $2$ holomorphic plane with positive end at $\gamma$.

The above configuration would appear to mess up the proof that $\partial^2=0$, because the above configuration cannot be interpreted in any obvious way as contributing to $\partial^2=0$. However I claim that if $J$ is generic, then the above degeneration never happens.

The proof will use intersection theory, as in the definition of ECH and the work of Siefring.

4. Why this degeneration never happens.

Suppose that $u$ is a holomorphic cylinder which is “close to breaking” into the holomorphic building $(u_1,u_2)$. To explain what I mean by this, let us not mod out by ${\mathbb R}$ translation on ${\mathbb R}\times Y$. Then there is some large positive real number $R$ and some small $\epsilon>0$ such that the intersection of $u$ with $[R,\infty)\times Y$ is within distance $\epsilon$ (pick your favorite norm) of the translate of $u_1$ by $+2R$, the intersection of $u$ with $[-R,R]\times Y$ is within distance $\epsilon$ of the $m$-fold cover of ${\mathbb R}\times\gamma$ union ${\mathbb R}\times\gamma$, and the intersection of $u$ with $(-\infty,R]\times Y$ is within distance of $\epsilon$ of the translate of $u_2$ by $-2R$.

Now let $C$ denote the intersection of $u$ with $[-2R+T,2R+T]\times Y$ where $T$ is a large constant which is independent of $R$. The curve $u$ is necessarily somewhere injective (because $u_2$ is), so $C$ is a surface which is embedded except for finitely many singularities. Let $\delta\ge 0$ denote the count of singularities of $C$ with the usual positive integer weights. Let $\zeta_+$ denote the intersection of $u$ with $\{2R+T\}\times Y$, regarded as a braid with $m+1$ strands in a three-dimensional tubular neighborhood of the Reeb orbit $\gamma$. Likewise let $\zeta_-$ denote the braid obtained by intersecting $u$ with $\{-2R+T\}\times Y$.

A version of the relative adjunction formula, cf. my lecture notes on ECH, section 3.3, implies that

$0 = -1 + w(\zeta_+) - w(\zeta_-) -2\delta.$

Here $w(\zeta_\pm)$ denotes the writhe of the braid $\zeta_\pm$ with respect to the usual trivialization. In particular, it follows that we have a strict inequality

$w(\zeta_+) > w(\zeta_-)$.

We are now going to obtain some independent bounds on the writhes $w(\zeta_\pm)$ and get a contradiction.

5. Writhe bounds.

First, the positive asymptotics of $u$ give the writhe bound

$w(\zeta_+) \le m\lfloor (m+1)\theta\rfloor,$

cf. the lecture notes on ECH, Lemma 5.5(a). This bound can be improved when $m+1$ and $\lfloor(m+1)\theta\rfloor$ have a common factor; one can then subtract $gcd(m+1,\lfloor (m+1)\theta\rfloor) - 1$ from the right hand side. I think this is proved in Siefring’s paper on intersection theory. However we will not need that improvement here.

Now the braid $\zeta_-$ has two components: a component $\zeta_-^1$ with one strand, and a component $\zeta_-^m$ with $m$ strands. Because the negative end of $u$ corresponding to $\zeta_-^m$ decays exponentially for time $4R-T$ before becoming $\zeta_-^m$, while the positive end of $u_2$ corresponding to $\zeta_-^1$ exponentially decays only for time $T$ before becoming $\zeta_-^1$, it follows that if $R$ is sufficiently large then the braid $\zeta_-^1$ wraps around the braid $\zeta_-^m$. Therefore

$w(\zeta_-) = w(\zeta_-^m) + 2m\eta(\zeta_-^1),$

where $\eta(\zeta_-^1)$ denotes the winding number of $\zeta_-^1$ around $\gamma$.

Again, as in Lemma 5.5(a) of the ECH lecture notes, the negative asymptotics of $u$ imply that

$w(\zeta_-^m)\ge (m-1)\lceil m\theta\rceil.$

(Again, this inequality can be improved when $m$ and $\lceil m\theta\rceil$ have a common factor, but we do not need this.) Finally, similarly to Proposition 3.2 in my second gluing paper with Taubes, if $J$ is generic then

$\eta(\zeta_-^1)=1.$

Putting this all together, we get

$w(\zeta_+) - w(\zeta_-) \le m\lfloor (m+1)\theta\rfloor - (m-1)\lceil m\theta\rceil - 2m.$

6. One last step.

To complete the proof, we need to use the assumption that $u_1$ has Fredholm index zero. The Fredholm index of $u_1$ is

$ind(u_1) = 1 + (2\lfloor (m+1)\theta \rfloor + 1) - (2\lfloor m\theta\rfloor + 1) - (2\lfloor\theta\rfloor + 1)$

$= 2(\lfloor (m+1)\theta\rfloor - \lceil m\theta\rceil).$

Putting the fact that this is zero into the previous inequality, we get

$w(\zeta_+) - w(\zeta_-) \le \lceil m\theta\rceil - 2m.$

Since $\theta < 2$, we have $\lceil m\theta\rceil \le 2m$. Thus $w(\zeta_+) - w(\zeta_-)\le 0$, which is the desired contradiction.