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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One Response to Lagrangian capacities and Ekeland-Hofer capacities

  1. In fact, one can also show that if X is a convex toric domain then

    lim_{k\to\infty}\frac{1}{k}c_k^{EH}(X) = t_0.

    The reason is that any convex toric domain can be sandwiched between a polydisk and an ellipsoid with the same value of t_0; so by the monotonicity of Ekeland-Hofer capacities, it is enough to check the above equation for ellipsoids and polydisks, which you can do using the known formulas for their E-H capacities.

    In this way one can prove the inequality

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

    for convex toric domains without ever talking about the CH capacities. However, if you want to try to prove the reverse inequality, then it still seems more promising to work with the CH capacities so that you can play with breaking of holomorpic curves etc.

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