A guest post by Dans C-G and P

[The following is a guest post by Dan Cristofaro-Gardiner and Dan Pomerleano. If anyone else is interested in contributing a guest post, please feel free to contact me. A blog is a good outlet for short or informal mathematical thoughts which might not have a place in a traditional publication, and guest posting is convenient if you are not yet ready to start your own blog. -M.H.]

What can we say about the minimum number of Reeb orbits?

The paper From one Reeb orbit to two showed that any Reeb flow on a closed contact three-manifold must have at least two closed orbits. While examples exist with exactly two orbits (e.g. irrational ellipsoids), there is no known example of a contact manifold that is not a lens space where the Reeb flow has finitely many closed orbits. It is therefore natural to try to refine this result under additional assumptions, and there has been interesting work in this direction by Hofer-Wysocki-Zehnder, Colin-Honda, Ginzburg-Gurel-Macarini, and others.

One example of such a refinement is a theorem of Hutchings and Taubes, which states that, for a nondegenerate contact form, the Reeb flow must have at least three distinct embedded Reeb orbits on any manifold that is not a lens space. It turns out that if the contact structure is not torsion, one can slightly improve on this result:

Proposition 1. Let (Y,\lambda) be a closed contact three-manifold, and let \xi be the contact structure for \lambda. Assume that c_1(\xi) is not torsion. Then the Reeb flow has at least three distinct embedded orbits. If \lambda is nondegenerate, then the Reeb flow has at least four distinct embedded orbits.

The proof of this proposition is given below. The arguments are similar to those in “From one Reeb orbit to two”, so this post may also be of interest to anyone curious about that paper.

1. Spectral invariants and a review of ECH

Our proof (as well as the proof in “From one Reeb orbit to two”) uses the “spectral invariants” defined by Hutchings in Quantitative embedded contact homology. To recall their definition, let us begin by stating some basic facts about ECH under the assumption that \lambda is nondegenerate. Fix a class \Gamma \in H_1(Y). The group ECH(Y,\lambda,\Gamma) is the homology of a chain complex ECC. This chain complex is generated by orbit sets \alpha = \lbrace (\alpha_i,m_i) \rbrace, where the \alpha_i are distinct embedded Reeb orbits, the m_i are positive integers, and the total homology class of \alpha is equal to \Gamma. The orbit sets are required to be admissible, which means that each m_i is equal to 1 when \alpha_i is hyperbolic. It is known that ECH is an invariant of the contact structure \xi (in fact, it is known that ECH is an invariant of the three-manifold, but we will not need this). Thus, the group ECH(Y,\xi,\Gamma) is well-defined.

Let \sigma be a nonzero class in ECH(Y,\xi,\Gamma). We can define invariants c_{\sigma}(\lambda) for any contact form \lambda in the contact structure \xi. This works as follows. An orbit set has a symplectic action defined by \mathcal{A}(\lbrace (\alpha_i,m_i) \rbrace) = \sum_i m_i \int_{\alpha_i} \lambda. If \lambda is nondegenerate, define c_{\sigma}(\lambda) to be the “minimum symplectic action” required to represent the class \sigma. If \lambda is degenerate, define c_{\sigma}(\lambda) = \lim_{n \to \infty} c_{\sigma}(\lambda_n), where \lambda_n are a sequence of nondegenerate contact forms converging in C^0 to \lambda. This works essentially because the c_{\sigma}(\cdot) behave like symplectic capacities: they satisfy monotonicity and scaling axioms which make c_{\sigma}(\lambda) in the degenerate case well-defined. For the details, see for example “Quantitative embedded contact homology”.

Here is the key fact that we need about spectral invariants:

Fact 2. Let (Y,\lambda) be a (possibly degenerate) contact manifold. Let \sigma \in ECH(Y,\xi,\Gamma). Then c_{\sigma}(\lambda)=\mathcal{A}(\alpha), where \alpha is some orbit set for \lambda with total homology class \Gamma. If \lambda is nondegenerate, then \alpha is admissible.

This is proved similarly to Lemma 3.1(a) in “From one Reeb orbit to two”. The proof in the degenerate case uses a standard compactness argument for Reeb orbits of bounded action.

The idea of the proof of the proposition is now to look at the spectral invariants associated to a certain sequence of classes with gradings tending to infinity. If there are too few Reeb orbits, we will find a contradiction with known facts about the asymptotics of these spectral invariants.

2. U-sequences

To make this precise, we now introduce the notion of a “U-sequence”. Recall that ECH comes equipped with a “U-map”, which is a degree -2 map defined by counting I=2 curves. Also recall that Taubes showed that there is a canonical isomorphism

ECH_*(Y,\lambda,\Gamma) \cong \widehat{HM}^{-*}(Y,s_{\xi} + PD(\Gamma)),

where \widehat{HM} denotes the Seiberg-Witten Floer cohomology defined by Kronheimer and Mrowka. The U-map agrees with an analogous structure on \widehat{HM} under this isomorphism.

Let \Gamma be a class in H_1(Y). If c_1(\xi) + 2PD(\Gamma) is torsion, then ECH(Y,\xi,\Gamma) has a relative \mathbb{Z} grading. It follows from the above isomorphism together with known facts about \widehat{HM} that this group is infinitely generated. In fact, it is well-known (by again using this isomorphism) that one can always find a U-sequence, namely a sequence of non-zero classes \sigma_k \in ECH(Y,\xi,\Gamma) with definite gradings such that U(\sigma_k) = \sigma_{k-1}. We will use a refined version of this statement, involving the canonical mod 2 grading on ECH (in this grading, the grading of an orbit set \alpha is (-1)^{h(\alpha)}, where h is the number of positive hyperbolic orbits in the orbit set).

Fact 3. Let (Y,\lambda) be a contact manifold. Assume that c_1(\xi) + 2PD(\Gamma) is torsion. Then either:

  • we have b_1(Y)=0, in which case there is a U-sequence in even grading, or
  • b_1(Y)>0, in which case there exist U-sequences in both even and odd grading.

This result can be deduced from the discussion in Section 35.1 of Kronheimer and Mrowka’s book “Monopoles and three-manifolds”.

3. A digression about odd contact manifolds

Fact 3 will be used in our proof of Proposition 1, but it has other interesting consequences as well. For example, let us say that a contact three-manifold (Y,\lambda) is “odd” if all closed embedded Reeb orbits are either elliptic or negative hyperbolic. It was asked previously on this blog whether all odd contact manifolds are lens spaces. Corollary 4 below provides some evidence in favour of this. If (Y,\lambda) is odd, then ECH(Y,\lambda,\Gamma) must be concentrated in even degree. We obtain as a corollary of Fact 3 that:

Corollary 4. If (Y,\lambda) is an odd contact manifold, then b_1(Y)=0.

4. The proof

Returning to the proof of Proposition 1, we will also need the following facts about the spectral invariants of a U-sequence associated to any contact form $\latex lambda$:

Fact 5.

  • Let \sigma be a nonzero class on ECH with U\sigma \ne 0. Then c_{U(\sigma)}(\lambda) < c_{\sigma}(\lambda).
  • Let \lbrace \sigma_k \rbrace be a U-sequence. Then

\lim_{k \to \infty} \frac{c_{\sigma_k}(\lambda)^2}{k} = 2vol(Y,\lambda).

The first item follows from Stokes’ Theorem in the nondegenerate case; when \lambda is degenerate, the key result is a compactness result for pseudoholomorphic currents due to Taubes, see “From one Reeb orbit to two”. The second item follows from the “volume conjecture” proved in “The asymptotics of embedded contact homology capacities”.

We have now laid out all of the necessary machinery to give our proof.

Proof.

The nondegenerate case. Suppose we have exactly three embedded orbits.

Our manifold (Y,\lambda) must not be odd in view of Corollary 4. We will next show that we must have exactly two elliptic orbits. Choose \Gamma such that c_1(\xi) + 2PD(\Gamma) is torsion. If we had zero elliptic orbits, it follows from the definition of the ECH chain complex that ECH(Y,\lambda,\Gamma) would be finitely generated, contradicting (for example) Fact 3. Let \lbrace \sigma_k \rbrace be a U-sequence. If we had one elliptic orbit e_1 and two hyperbolic orbits h_1,h_2, we would contradict Fact 5. More precisely, the first bullet of Fact 5 together with Fact 2 would imply that \frac{c_{\sigma_k}(\lambda)^2}{k} would have to grow at least linearly with k, while the second bullet implies that this cannot occur.

Thus, we can assume that we have two elliptic orbits e_1,e_2 and a positive hyperbolic h.

[There can’t be three elliptic orbits because this would contradict Theorem 1.2 in The Weinstein conjecture for stable Hamiltonian structures. -Ed.] The key fact is now that since c_1(\xi) is not torsion, \Gamma is also not torsion. The significance of this is as follows. We have an induced map \mathbb{Z}^2= \mathbb{Z}[e_1] \oplus \mathbb{Z}[e_2] \to H_1(Y), which sends [e_i] to the class represented by the Reeb orbit e_i. If the kernel has rank zero, then again, ECH(Y,\lambda,\Gamma) would be finitely generated. If the kernel has rank two, then these orbits would represent torsion classes in homology. On the other hand, by Fact 3, we must have a U-sequence in ECH(Y,\lambda,\Gamma) in even degree. This must take the form e_1^{m_k}e_2^{n_k}, contradicting our assumption that c_1(\xi) is non-torsion.

It remains to handle the case when the kernel has rank one. In this case, assume that the kernel is generated by some integer vector (c,d), say with d>0. Then each generator of our U-sequence e_1^{m_k}e_2^{n_k} must have the form e_1^{m_0+x_kc}e_2^{n_0+x_kd}. Because there are infinitely many distinct e_1^{m_k}e_2^{n_k}, we must have c\ge 0 (otherwise we would have -n_0\le x_k \le m_0 for all k). Since c and d are nonnegative, the asymptotics of this sequence would again violate the second bullet of Fact 5, since the action of each term in this sequence would have to be bigger than the action of the previous term by at least the minimum of the actions of e_1 and e_2.

The degenerate case. By “From one Reeb orbit to two”, we have at least two distinct embedded Reeb orbits. So assume that we have exactly two, \gamma_1 and \gamma_2. We now argue similarly to before. Namely, again consider the U-sequence latex $\lbrace \sigma_k \rbrace$, as well as the induced map \mathbb{Z}^2= \mathbb{Z}[\gamma_1] \oplus \mathbb{Z}[\gamma_2] \to H_1(Y). By Fact 2, this kernel cannot have rank two. By Fact 2, and the first bullet point of Fact 5, the kernel does not have rank 0. By repeating the argument in the previous paragraph, it also cannot have rank 1.

QED

[Any ideas for improving the above bounds further? As suggested at the beginning, one might conjecture that if (Y,\lambda) is a closed contact three-manifold, and if Y is not (a sphere or) a lens space, then there are infinitely many Reeb orbits. -Ed.]

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Posted in ECH, Open questions | 1 Comment

Lagrangian capacities and Ekeland-Hofer capacities

References for this post:

[CM] K. Cieliebak and K. Mohnke, Punctured holomorphic curves and Lagrangian embeddings

[HN] M. Hutchings and J. Nelson, Cylindrical contact homology for dynamically convex contact forms in three dimensions

[BEYOND] M. Hutchings, Beyond ECH capacities

[HL] R. Hind and S. Lisi, Symplectic embeddings of polydisks

First, let me mention that Chris Wendl has a new blog.

Now I would like to comment on one remark in the recent preprint [CM] (which has many more interesting things in it). In this paper, Cieliebak and Mohnke define the “Lagrangian capacity” of a symplectic manifold (X,\omega) as follows: If T\subset X is a Lagrangian torus, define A_{min}(T) to be the infimum of \int_D\omega where D\in\pi_2(X,T) and \int_D\omega>0. Then define c_L(X,\omega) to be the supremum of A_{min}(T) over all embedded Lagrangian tori T. Cieliebak-Mohnke then ask:

Question. For which domains X in {\mathbb R}^{2n} is it true that

c_L(X) = \lim_{k\to\infty}\frac{1}{k} c_k^{EH}(X)

where c_k^{EH} denotes the k^{th} Ekeland-Hofer capacity?

They conjecture that this is true for ellipsoids and ask whether it is true for all convex domains.

I would now like to present some evidence (based on more conjectures) that the answer to the Question is YES for an interesting family of examples, namely convex toric domains in {\mathbb R}^4. More precisely, what I will do is the following:

  • Recall how to use cylindrical contact homology to define an ersatz version of the Ekeland-Hofer capacities, denoted by c_k^{CH}, which are conjecturally equal to them.
  • Compute c_k^{CH} for convex toric domains in {\mathbb R}^4, modulo a conjectural description of the cylindrical contact homology differential which is probably not too hard to prove.
  • Deduce from the above computation that lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X) \le c_L(X) whenever X is a convex toric domain in {\mathbb R}^4.
  • Briefly discuss strategy for trying to prove the reverse inequality.

1. Cylindrical contact homology capacities

Let X be a (strictly) star-shaped domain in {\mathbb R}^4 with boundary Y. Recall that the Liouville form

\lambda = \frac{1}{2}\sum_{i=1}^2(x_idy_i-y_idx_i)

restricts to a contact form on Y. Let us perturb Y if necessary to ensure that this contact form is nondegenerate, and let us further assume that \lambda|_Y is dynamically convex (which holds for example when X is convex). We can then define the cylindrical contact homology CH(Y,\lambda), as explained in [HN]. (The proof of invariance of cylindrical contact homology in the dynamically convex case and construction of cobordism maps on it are to appear in a sequel.) With the usual grading convention, this cylindrical contact homology is {\mathbb Q} in degree 2,4,\ldots and 0 in all other degrees.

If k is a positive integer, we now define c_k^{CH}(X) to the minimum over L such that the degree 2k class in CH(Y,\lambda) can be represented by a linear combination of good Reeb orbits, each of which has action \le L. One can use cobordism maps to show that this number does not depend on the almost complex structure used to define CH(Y,\lambda) and is monotone with respect to symplectic embeddings. Also, this definition extends to any convex domain X (where the boundary might not be smooth or nondegenerate) by taking C^0 limits.

It is conjectured that c_k^{CH}(X)=c_k^{EH}(X) when X is a convex domain in {\mathbb R}^4, or more generally a star-shaped domain whose boundary is the limit of hypersurfaces which are nondegenerate and dynamically convex. I made (a more general version of) this conjecture in this previous post, based on calculations for ellipsoids and polydisks (which I will explain below), and other people have made similar conjectures.

2. CH capacities of convex toric domains in {\mathbb R}^4

Recall that if \Omega is a domain in the first quadrant in the plane, we define the “toric domain”

X_\Omega = \{z\in{\mathbb C}^2 \mid \pi(|z_1|^2,|z_2|^2)\in\Omega\}.

I’ll use the not entirely satisfactory term “convex toric domain” to indicate a domain X_\Omega for which

\Omega = \{(x,y)\mid 0\le x\le a,\; 0\le y \le f(x)\}

where f:[0,a]\to[0,\infty) is a nonincreasing concave function. Let’s now compute c_k^{CH}(X_\Omega) where X_\Omega is a convex toric domain.

As explained in [BEYOND], the boundary Y of X_\Omega can pe perturbed so that the contact form is nondegenerate and, up to large symplectic action, the simple Reeb orbits consist of the following:

  • Elliptic orbits e_{1,0} and e_{0,1}. Here e_{1,0} is the circle in Y where z_1=0, and e_{0,1} is the circle in Y where z_2=0.
  • An elliptic orbit e_{a,b}, and a hyperbolic orbit h_{a,b}, for each pair (a,b) of relatively prime positive integers. These arise from points on the boundary of \Omega where the slope of a tangent line to \partial\Omega is -b/a.

If (a,b) are nonnegative integers (not both zero), let d denote their greatest common divisor, and let a'=a/d and b'=b/d. Let e_{a,b} denote the d-fold cover of e_{a',b'}, and let h_{a,b} denote the d-fold cover of h_{a',b'} (when a and b are both nonzero). The generators of the cylindrical contact homology then consist of the following:

  • e_{a,b} and h_{a,b} where a,b are positive integers.
  • e_{d,0} and e_{0,d} where d is a positive integer.

It follows from calculations in [BEYOND] that the gradings of these generators are given by

|e_{a,b}| = 2(a+b),

|h_{a,b}|=2(a+b)-1.

Based on ECH calculations, I think the following should not be too hard to prove:

Conjecture. For a suitable generic J, the differential on the cylindrical contact homology chain complex is given by

\partial e_{a,b}=0,

\partial h_{a,b} = d(\pm e_{a-1,b} \pm e_{a,b-1})

where d denotes the greatest common divisor of a and b.

If you believe this, then it follows that the degree 2k homology generator is represented by e_{a,b} with a+b=k, and these are all homologous. Thus c_k^{CH}(X_\Omega) is the minimum of the symplectic action of e_{a,b} where a+b=k.

What is this symplectic action? The calculations in [BEYOND] show that, up to some small error coming from the perturbation, the symplectic action of e_{a,b} is given by

A_\Omega(e_{a,b}) = (a,-b)\times p_{\Omega,-b/a}

where p_{\Omega,-b/a} denotes a point on \partial\Omega where a tangent line to \partial\Omega has slope -b/a, and \times denotes the cross product. An equivalent way to say this, which is a bit more convenient for the present calculation, is

A_\Omega(e_{a,b}) = \max\{ bx+ay \mid (x,y)\in\Omega\}.

We conclude that

c_k^{CH}(X_\Omega) = \min_{a+b=k}\max\{bx+ay\mid (x,y)\in\Omega\}

where the minimum is over nonnegative integers a,b.

3. Examples of CH capacities

To become more comfortable with the above formula, let us compute some examples of CH capacities and check that they agree with the known formulas for ECH capacities.

First suppose that \Omega is the rectangle with vertices (0,0), (c,0), (0,d), (c,d) so that X_\Omega is the polydisk P(c,d). Then

c_k^{CH}(P(c,d)) = \min_{a+b=k}(bc+ad) = k\max(c,d).

This agrees with Ekeland-Hofer.

Next suppose that \Omega is the triangle with vertices (0,0), (c,0), (0,c) so that X_\Omega is the ball B(c). Then

c_k^{CH}(B(c)) = \min_{a+b=k}\max(ac,bc) = c\lceil k/2 \rceil

which also agrees with Ekeland-Hofer. Finally, one can generalize this to compute c_k^{CH} of an ellipsoid and check that it agrees with Ekeland-Hofer. (The Ekeland-Hofer capacities of the ellipsoid E(c,d) consist of the positive integer multiples of c and d, arranged in nondecreasing order.) But I’ll skip this since it is an unnecessarily complicated way to compute the CH capacities of an ellipsoid. (It is much easier to just take a standard irrational ellipsoid with exactly two simple Reeb orbits.)

4. Comparison with the Lagrangian capacity

Let t_0 denote the largest positive real number t such that (t,t)\in\Omega. I claim that

c_L(X_\Omega)\ge t_0

and

\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega)=t_0.

The first claim is easy, because if (t,t)\in\Omega, then the torus T=(\pi|z_1|^2 = \pi|z_2|^2=t) is a Lagrangian torus in X_\Omega such that A_{min}(T)=t.

To prove the second claim, note that by Part 2, we have

\lim_{k\to\infty}\frac{1}{k} c_k^{CH}(X_\Omega) = \min_{a+b=1}\max\{bx+ay\mid (x,y)\in\Omega\}.

Here a,b are now nonnegative real numbers instead of integers.

If a+b=1, then taking (x,y)=(t_0,t_0) shows that \max\{bx+ay\mid (x,y)\in\Omega\}\ge t_0, and thus

\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega) \ge t_0.

To prove the reverse inequality, consider a tangent line to \partial\Omega through the point (t_0,t_0). We can uniquely write this line in the form bx+ay=L where a+b=1. Since this line is tangent to \partial\Omega, we have \max\{bx+ay\mid(x,y)\in\Omega\}= L, and thus

\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega)\le L.

On the other hand, since the line goes through the point (t_0,t_0), we have

L = bt_0 + at_0 = t_0.

It follows that

\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega) \le t_0.

This completes the proof of the claims. We conclude that

\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X_\Omega) \le c_L(X_\Omega).

5. How to prove the reverse inequality?

Now we would like to prove the reverse inequality

\lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X) \ge c_L(X).

where X=X_\Omega (and here it is maybe not so important that X is a convex toric domain). To do so, let T\subset X be an embedded Lagrangian torus. We want to prove that there exists D\in\pi_2(X,T) such that

0 < \int_D\omega \le \lim_{k\to\infty}\frac{1}{k}c_k^{CH}(X).

(Actually, in this case, since \pi_2(X,T)=H_2(X,T), we could allow D to be any (not necessarily embedded) compact oriented surface in X with boundary on T.)
I haven’t thought this through, but maybe one prove this using the methods in [HL]. Or maybe these methods will just prove the following weaker upper bound?

Namely, [CM,Cor. 1.3] and monotonicity of the Lagrangian capacity imply the upper bound

c_L(X_\Omega) \le \frac{1}{2}\max\{x+y\mid (x,y)\in\Omega)\}.

This agrees with our trivial lower bound c_L(X_\Omega)\le t_0 if and only if a tangent line to \partial\Omega through (t_0,t_0) has slope -1; or equivalently, P(c,c)\subset X_\Omega \subset B(2c) for some c.

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Version 2 of “Beyond ECH capacities”

I just posted a revised version of the preprint “Beyond ECH capacities” to the arXiv. It should appear on Monday, but you can view it here first. The new version corrects some embarrassing/horrifying typos, clarifies a few things, and includes the new application to sharpness of symplectic folding from the previous blog post.

There are still tons of calculations to do using the ideas in this preprint, if anyone is interested. Here are some examples of things to try:

  • Study symplectic embeddings of the polydisk P(a,1) into a ball when a>12/5, improving Theorem 1.3 and/or extending Theorem 1.4 in the preprint.
  • Study symplectic embeddings of the polydisk P(a,1) into the ellipsoid P(bc,c) when b is an integer and a>2, or when b is not an integer, extending Theorem 1.5 in the preprint.
  • Prove Conjecture A.3 in the preprint (regarding the ECH differential on the boundary of a suitably perturbed convex toric domain), which would allow Theorem 1.6 to be improved as explained in Remark 1.8.
  • Study symplectic embeddings of the disjoint union of two polydisks into an ellipsoid or polydisk.
  • Study symplectic embeddings of concave or convex toric domains into a concave toric domain.

In general, one would like to identify more cases when the obvious inclusion map, or the folding-type constructions in Felix Schlenk’s book, give optimal symplectic embeddings.

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Symplectic folding is sometimes optimal

Reference: [BEYOND] = “Beyond ECH capacities”

I have played with a few more calculations using the methods in [BEYOND]. Here is the most interesting thing I have found so far.

In [BEYOND, Thm. 1.2], it was shown, among other things, that if the polydisk P(a,1) symplectically embeds into the four-dimensional ball B(c), and if 2 \le a \le 4, then c\ge (10+a)/4. On the other hand, Felix Schlenk showed using symplectic folding that if 2\le a\le 4, then P(a,1) symplectically embeds into B(c) whenever c> (4+a)/2. These bounds agree for a=2 and disagree for a>2. One might ask whether one can improve one or both of these bounds to get them to agree. In fact, it turns out that symplectic folding is sometimes optimal, in the following sense:

Theorem. If 2\le a\le 12/5, and if P(a,1) symplectically embeds into B(c), then c\ge (4+a)/2.

This is a direct application of [BEYOND, Thm. 1.18], and I will assume the statement of the latter theorem below. (In other words, the following is basically an addendum to be added in the next version of [BEYOND], unless I discover some way to improve it first.)

Proof of Theorem. Suppose that 2\le a\le 12/5, that P(a,1) symplectically embeds into B(c), and that c < (4+a)/2. We will obtain a contradiction in four steps. Below, the symbol \le between convex generators means \le_{P(a,1),B(c)}.

Step 1. We first show that if \Lambda \le e_{1,1}^d with d\le 9, then y(\Lambda) \le 1.

If y(\Lambda)\ge 2, then as in Step 1 of the proof of [BEYOND, Thm. 1.2], we have

3d-3+2a \le dc.

Combining this with our assumption that c < (4+a)/2 gives

(d-4)a > 2d-6.

If d<4 then it follows that a < 4/3; if d=4 then it follows that 2<0; and if 5\le d\le 9 then it follows that a>12/5. Either way this contradicts our hypothesis that 2\le a \le 12/5.

Step 2. We now show that if \Lambda\le e_{d,d}, and if y(\Lambda)\le 1, then \Lambda includes a factor of e_{1,0}.

If not, then the only possibility for \Lambda with the correct ECH index is

\Lambda = e_{(d^2+3d-2)/2,1}.

The action inequality in the definition of \le_{P(a,1),B(c)} then implies that

(d^2+3d-2)/2 + a \le dc.

Combining this with our assumption that c < (4+a)/2 gives

(d-2)a > d^2-d-2.

If d=1 then it follows that a<2; if d=2 then it follows that 0<0; and if d\ge 3 then it follows that a > d+1. Either way this contradicts our hypothesis that 2 \le a \le 12/5.

Step 3. We now show that there does not exist any convex generator \Lambda with \Lambda \le e_{1,1}^9.

If \Lambda is such a generator, then we know from Step 1 that y(\Lambda) \le 1.

If y(\Lambda) = 0, then the only possibility for \Lambda with the correct ECH index is \Lambda = e_{1,0}^{54}. Then 54 \le 9c, which combined with our assumption that c < (4+a)/2 implies that a > 8, contradicting our hypotheses.

If y(\Lambda) = 1, then x(\Lambda) \ge 27, or else we would have I(\Lambda) \le 106, contradicting the fact that I(\Lambda) = 108. Since x(\Lambda) \ge 27, it follows that

27 + a \le 9c.

Combining this with our assumption that c < (4+a)/2 gives a > 18/7, contradicting our hypothesis that a \le 12/5.

Step 4. We now apply [BEYOND, Thm. 1.18] to \Lambda'=e_{1,1}^9, to obtain a convex generator \Lambda, and factorizations \Lambda=\Lambda_1\cdots\Lambda_n and \Lambda'=\Lambda_1'\cdots\Lambda_n', satisfying the three bullet points in [BEYOND, Thm. 1.18].

By Step 3 and the first bullet point, we must have n>1.

By Step 2 and the first two bullet points, all of the \Lambda_i must be equal, and all of the \Lambda_i' must be equal. Thus either n=9 and \Lambda_i'=e_{1,1} for each i, or n=3 and \Lambda_i'=e_{3,3} for each i.

If n=9, then by Steps 1 and 2, we have \Lambda=e_{1,0}^2 for each i. But then I(\Lambda)=36, contradicting the fact that I(\Lambda)=108.

If n=3, then by Step 1, and the facts that I(\Lambda_i)=18 and x(\Lambda_i) + y(\Lambda_i) \ge 8, the only possibilities are that \Lambda_i = e_{1,0}^9 for each i, or \Lambda_i = e_{1,0}e_{6,1} for each i. In the former case we have I(\Lambda) = 54, and in the latter case we have I(\Lambda)=102. Either way, this contradicts the fact that I(\Lambda)=108.

QED

Remark. It is conceivable that with more work, the hypothesis a\le 12/5 could be weakened to a \le (\sqrt{7}-1)/(\sqrt{7}-2) = 2.54858\cdots. The significance of the latter number is that if a is less than it, and if d is sufficiently large with respect to a, then there does not exist any convex generator \Lambda with \Lambda \le e_{1,1}^d. We might then be able to use arguments similar to the above to get a contradiction.

More generally, one can maybe get more information by considering all of the holomorphic curves that exist in the cobordism coming from a symplectic embedding. We know that certain curves must exist in order to give a chain map on ECH satisfying the required properties. However the existence of certain curves excludes the existence of others, for example when their intersection number would be negative. When you write down all of these conditions, it is like a giant logic puzzle, and the challenge is to extract significant information from it using a manageable amount of computation.

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Comparison with Ekeland-Hofer

Helmut Hofer asked me how the symplectic embedding obstructions in “Beyond ECH capacities” compare to the obstructions given by Ekeland-Hofer capacities, for symplectic embeddings between ellipsoids and polydisks. I happened to know the answer to this, using something which I chose to leave out of the paper (too many things going on, needed to try to stay focused!), so I would like to explain it here. First, the answer is the following:

  • For symplectic embeddings of (four-dimensional) ellipsoids into ellipsoids or polydisks, ECH capacities give sharp obstructions (shown by McDuff), while Ekeland-Hofer capacities are often weaker.
  • For a polydisk P(a,b) into another polydisk P(a',b') where a\ge b and a'\ge b', Ekeland-Hofer only tells us that b\le b'. ECH capacities say a bit more but are not very good. “Beyond ECH capacities” gives sharp obstructions in some cases.
  • For a polydisk P(a,b) into an ellipsoid E(c,d), ECH capacities are not very good, and sometimes weaker than Ekeland-Hofer capacities. Ekeland-Hofer gives a sharp obstruction when a=b (and a less good obstruction when a\neq b). “Beyond ECH capacities” can recover this (the sharp obstruction when a=b; I haven’t checked that I can recover all information given by all Ekeland-Hofer capacities when a\neq b).

I would now like to explain this last point. First let’s change the notation: given a,b\ge 1, we would like to find the infimum of c such that the polydisk P(a,1) symplectically embeds into the ellipsoid E(bc,c). Observe that P(a,1) trivially embeds into E(bc,c) by inclusion if c\ge 1+a/b. When a=1, the converse is true:

Theorem 1. If b\ge 1 and P(1,1) symplectically embeds into E(bc,c), then c\ge 1+1/b.

Here’s how to prove this using Ekeland-Hofer capacities. Let’s denote the k^{th} Ekeland-Hofer capacity by c_k. (Usually I use this symbol to denote ECH capacities, but we won’t be talking about ECH capacities in this post.) The Ekeland-Hofer capacities of P(a,1) for a\ge 1 are given by

c_k(P(a,1)) = k.

On the other hand, c_k(E(bc,c)) is the k^{th} entry in the list of all positive integer multiples of bc or c, written in increasing order with repetitions. It follows that

c_{k + \lfloor k/b \rfloor}(E(bc,c)) = kc.

So if P(1,1) symplectically embeds into E(bc,c), then by the monotonicity property of Ekeland-Hofer capacities, for every positive integer k we have

k + \lfloor k/b \rfloor \le kc.

Taking k large gives 1 + 1/b \le c as desired. Now how do we recover this from “Beyond ECH capacities”? Doing something rather crude with the methods in that paper, which I will explain below, one obtains the following:

Theorem 2. Suppose P(a,1) symplectically embeds into P(bc,c) where a,b\ge 1. Suppose also that

\sqrt{a/2} + \sqrt{1/(2a)} \le \sqrt{b} + 1/\sqrt{b}.

Then

2c \ge 1 + b^{-1} + \sqrt{1 + b^{-2}} + a(1 + b^{-1} - \sqrt{1 + b^{-2}}).

For example, if a=1, then we obtain c \ge 1 + b^{-1}, recovering Theorem 1.  Another example is that if b=1, then we obtain

c \ge (1 + 1/\sqrt{2}) + (1-1/\sqrt{2})a

for a\le 3+2\sqrt{2}. This is nontrivial, but weaker than Theorem 1.2 in “Beyond ECH capacities” when a>1.

Now I will explain the proof of Theorem 2, assuming as a prerequisite the statement of Theorem 1.18 in “Beyond ECH capacities”. We will need the following immediate corollary of the latter theorem:

Theorem 3. Let X_\Omega and X_{\Omega'} be convex toric tomains, and suppose that X_\Omega symplectically embeds into X_{\Omega'}. Let \Lambda' be a convex generator which is minimal for X_{\Omega'}. Then there exists a convex generator \Lambda such that

I(\Lambda) = I(\Lambda'),

A_{\Omega}(\Lambda) \le A_{\Omega'}(\Lambda'),

and

x(\Lambda) + y(\Lambda) \ge x(\Lambda') + y(\Lambda').

(To address a question of Dan C-G: You can also say that I - x - y defines a filtration on the ECH chain complex which is preserved by the cobordism map. However I’m not sure if the cobordism map necessarily induces an isomorphism on the homology of the associated graded, because I don’t know if this filtration will be preserved by the relevant chain homotopies. There are also variants of this filtration to play with coming from J_0 and J_+. An interesting topic to think about later.)

Here is how to deduce Theorem 2 from Theorem 3. Let \Lambda' be a convex generator which is minimal for E(bc,c) and has very large ECH index. Let us rescale this so that the area under the curve is b/2. Then the rescaled \Lambda' is approximately a straight line from (0,1) to (b,0). Now the convex generator \Lambda provided by Theorem 2, after rescaling by the same factor, is a curve from (0,y) to (x,0), for some positive real numbers x and y, which is the graph of a nonincreasing concave function, and the area under this curve is approximately b/2. It follows that

xy/2 \le b/2 \le xy,

up to some error which can be made arbitrarily small by taking the ECH index of \Lambda to be sufficiently large. The other two inequalities in Theorem 2 then tell us that, also up to a small error, we have

x+ay \le bc

and

x+y \ge b+1.

It follows that

bc\ge \min\{x+ay | xy/2\le b/2 \le xy, x+y\ge b+1\}.

It is now an exercise in undergraduate multivariable calculus to compute the minimum on the right hand side. If

\sqrt{a/2} + \sqrt{1/(2a)} \ge \sqrt{b} + 1/\sqrt{b},

then the minimum is \sqrt{2ab}. Thus we conclude that c\ge \sqrt{2a/b}. This is just the volume constraint vol(P(a,1)) \ge vol(E(bc,c)). On the other hand, if

\sqrt{a/2} + 1/\sqrt{1/(2a)} \ge \sqrt{b} + 1/\sqrt{b},

then the minimum is (b+1+\sqrt{b^2+1})/2 + a(b+1-\sqrt{b^2+1})/2, which proves Theorem 3.

One last remark: the asymptotics of the symplectic embedding obstructions coming from ECH capacities for large ECH index just recover the volume constraint. The example above shows that the asymptotics of the obstruction in Theorem 2 for large ECH index (and also the Ekeland-Hofer capacities for large k) sometimes say more.      

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Beyond ECH capacities

In case the last couple of postings were confusing, I have now posted a preprint explaining the story. Here is the abstract:

“ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain and target are ellipsoids (proved by McDuff), and more generally when the domain is a “concave toric domain” and the target is a “convex toric domain” (proved by Cristofaro-Gardiner). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. This paper uses more refined information from ECH to give stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind-Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.”

There are lots more calculations to do, to try to use the techniques in this paper to obstruct more symplectic embeddings. If anyone is interested in working on some, please feel free to discuss this with me.

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Symplectic embeddings of polydisks into polydisks

If I am not mistaken, the methods in the previous post (plus a conjecture about the ECH chain complex of perturbed boundaries of convex toric domains) can be used to show that if a,b,c are real numbers with a,b\ge 1 and c>0, and if P(a,1) symplectically embeds into P(bc,c), and if a\le 2b, then a\le bc. In other words, if you include one four-dimensional polydisk into another, such that the long sides are the same, and the short side of the domain is at least half the short side of the target, then this symplectic embedding is optimal.

The conjecture needed is that in the ECH chain complex of the (perturbed) boundary of a convex toric domain, a generator with only elliptic orbits represents a nontrivial homology class. (This would follow from a conjectural description of the differential in terms of rounding corners.) Without this conjecture, one can still prove a version of the above theorem in which the hypothesis a\le 2b is strengthened somewhat. (When b=1 you can still just assume a\le 2.)

I’m working on writing this up cleanly.

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