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 as follows: If is a Lagrangian torus, define to be the infimum of where and . Then define to be the supremum of over all embedded Lagrangian tori . Cieliebak-Mohnke then ask:
Question. For which domains in is it true that
where denotes the 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 . 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 , which are conjecturally equal to them.
- Compute for convex toric domains in , modulo a conjectural description of the cylindrical contact homology differential which is probably not too hard to prove.
- Deduce from the above computation that whenever is a convex toric domain in .
- Briefly discuss strategy for trying to prove the reverse inequality.
1. Cylindrical contact homology capacities
Let be a (strictly) star-shaped domain in with boundary . Recall that the Liouville form
restricts to a contact form on . Let us perturb if necessary to ensure that this contact form is nondegenerate, and let us further assume that is dynamically convex (which holds for example when is convex). We can then define the cylindrical contact homology , 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 in degree and in all other degrees.
If is a positive integer, we now define to the minimum over such that the degree class in can be represented by a linear combination of good Reeb orbits, each of which has action . One can use cobordism maps to show that this number does not depend on the almost complex structure used to define and is monotone with respect to symplectic embeddings. Also, this definition extends to any convex domain (where the boundary might not be smooth or nondegenerate) by taking limits.
It is conjectured that when is a convex domain in , 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
Recall that if is a domain in the first quadrant in the plane, we define the “toric domain”
I’ll use the not entirely satisfactory term “convex toric domain” to indicate a domain for which
where is a nonincreasing concave function. Let’s now compute where is a convex toric domain.
As explained in [BEYOND], the boundary of 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 and . Here is the circle in where , and is the circle in where .
- An elliptic orbit , and a hyperbolic orbit , for each pair of relatively prime positive integers. These arise from points on the boundary of where the slope of a tangent line to is .
If are nonnegative integers (not both zero), let denote their greatest common divisor, and let and . Let denote the -fold cover of , and let denote the -fold cover of (when and are both nonzero). The generators of the cylindrical contact homology then consist of the following:
- and where are positive integers.
- and where is a positive integer.
It follows from calculations in [BEYOND] that the gradings of these generators are given by
Based on ECH calculations, I think the following should not be too hard to prove:
Conjecture. For a suitable generic , the differential on the cylindrical contact homology chain complex is given by
where denotes the greatest common divisor of and .
If you believe this, then it follows that the degree homology generator is represented by with , and these are all homologous. Thus is the minimum of the symplectic action of where .
What is this symplectic action? The calculations in [BEYOND] show that, up to some small error coming from the perturbation, the symplectic action of is given by
where denotes a point on where a tangent line to has slope , and denotes the cross product. An equivalent way to say this, which is a bit more convenient for the present calculation, is
We conclude that
where the minimum is over nonnegative integers .
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 is the rectangle with vertices so that is the polydisk . Then
This agrees with Ekeland-Hofer.
Next suppose that is the triangle with vertices so that is the ball . Then
which also agrees with Ekeland-Hofer. Finally, one can generalize this to compute of an ellipsoid and check that it agrees with Ekeland-Hofer. (The Ekeland-Hofer capacities of the ellipsoid consist of the positive integer multiples of and , 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 denote the largest positive real number such that . I claim that
The first claim is easy, because if , then the torus is a Lagrangian torus in such that .
To prove the second claim, note that by Part 2, we have
Here are now nonnegative real numbers instead of integers.
If , then taking shows that , and thus
To prove the reverse inequality, consider a tangent line to through the point . We can uniquely write this line in the form where . Since this line is tangent to , we have , and thus
On the other hand, since the line goes through the point , we have
It follows that
This completes the proof of the claims. We conclude that
5. How to prove the reverse inequality?
Now we would like to prove the reverse inequality
where (and here it is maybe not so important that is a convex toric domain). To do so, let be an embedded Lagrangian torus. We want to prove that there exists such that
(Actually, in this case, since , we could allow to be any (not necessarily embedded) compact oriented surface in with boundary on .)
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
This agrees with our trivial lower bound if and only if a tangent line to through has slope ; or equivalently, for some .