Symplectic embeddings and the lagrangian bidisk

Symplectic embeddings and the lagrangian bidisk

Vinicius Gripp Barros Ramos
Abstract

In this paper we obtain sharp obstructions to the symplectic embedding of the lagrangian bidisk into four-dimensional balls, ellipsoids and symplectic polydisks. We prove, in fact, that the interior of the lagrangian bidisk is symplectomorphic to a concave toric domain using ideas that come from billiards on a round disk. In particular, we answer a question of Ostrover [12]. We also obtain sharp obstructions to some embeddings of ellipsoids into the lagrangian bidisk.

1 Introduction

Symplectic embedding questions have been central in the study of symplectic manifolds. The first of such questions was studied by Gromov in [6]. After that, many techniques were created to deal with the questions of when symplectic embeddings exist. Symplectic capacities are one of such techniques and they provide an obstruction to the existence of a symplectic embedding. An interesting general question is whether a certain capacity is sharp for a certain embedding problem, i.e., whether the symplectic embedding exists if and only if this capacity does not give an obstruction to it.

ECH capacities are a sequence of capacities which are defined for four-dimensional symplectic manifolds [7]. For a symplectic manifold , there is a sequence of real numbers:

These numbers satisfy the following properties:

  1. If , then , for every .

  2. If symplectically embeds into , then

  3. (1)

ECH capacities have been computed for many manifolds and they have been shown to be sharp for several symplectic embedding questions in dimension 4, see for example [10, 4, 5]. We recall that ECH capacities are said to be sharp for a certain embedding problem if

In this paper the symbol will always denote a symplectic embedding. The goal of this paper is to prove some new results concerning the symplectic embeddings of the lagrangian bidisk into four-dimensional balls, ellipsoids and polydisks. In particular, this answers a question of Yaron Ostrover in [12, §5].

We will now set up our notation. We will always consider with coordinates and its subsets endowed with the symplectic form

The main domain we are interested in is the lagrangian bidisk in , denoted by , which is defined to be

We observe that the lagrangian product of any two disks is symplectomorphic to a multiple of . We now define the ellipsoids and the symplectic polydisks as follows.

We denote the Euclidean ball of radius by .

The main result of this paper is the following theorem.

Theorem 1.

ECH capacities give a sharp obstruction to symplectically embedding the interior of into balls, ellipsoids and symplectic polydisks. Moreover,

  1. if and only if ,

  2. if and only if and ,

  3. if and only if .

Remark 2.

Part (c) of Theorem 1 was previously known by [15]. The proof combines the explicit construction of an embedding and Gromov’s non-squeezing theorem.

1.1 Toric domains

Although understanding symplectic embeddings of four-dimensional symplectic manifolds in general is a very hard problem, many results are known for a certain class of manifolds called toric domains, which are constructed as follows. If is a closed region in the first quadrant of , we define the toric domain to be

We endow with the restriction of the standard symplectic form in .

The main result needed to prove Theorem 1 is the following theorem.

Theorem 3.

Let be the toric domain , where is the region bounded by the coordinate axes and the curve parametrized by

(2)

Then and are symplectomorphic.

Remark 4.

The curve (2) has some nice properties. For example, if we switch by , we deduce that this curve is symmetric with respect to the reflection about the line . We also observe that

Two kinds of toric domains are of particular interest. Let be the domain in the first quadrant of bounded by the coordinate axes and a curve which is the union of the graph of a piecewise smooth non-increasing function and the line segment connecting and . We always assume that . If , we take . We say that is convex111This definition of convex toric domains is slightly less general than that given in [5], but it suffices for all of our applications. is is a concave function, and that is concave if is convex function and , see Figure 1(a,b). We observe that ellipsoids and symplectic polydisks are convex toric domains and that ellipsoids are the only toric domains that are both convex and concave. Moreover the toric domain defined in Theorem 3 is concave, see Figure 1(c). In [5], Cristofaro-Gardiner proved the following theorem.

Theorem 5 (Cristofaro-Gardiner).

Let and be concave and convex toric domains, respectively. Then ECH capacities give a sharp obstruction for embedding into .

We observe that the first claim of Theorem 1 follows immediately from Theorems 3 and 5.

(a) A convex toric domain
(b) A concave toric domain
(c) The region
Figure 1: Toric domains

1.2 The boundary of and billiards

The idea of the proof of Theorem 3 is to put an appropriate Hamiltonian toric action on and to compute the image of its moment map. We will now give a description of this action.

We would like first to define a toric action on and then to extend it to all of . We cannot do that because is not smooth. But we can still get an idea of the actual definition which will be given in §2.3 by looking at . The 3-manifold is a union of two solid tori . The characteristic flow is generated by the vector field defined by:

Note that we cannot extend continuously on . Even still, generates a continuous flow on so that each time we hit the torus , we go from the interior of a solid torus to another, and so that there are two orbits contained in which rotate along and with the same speed in the clockwise and counter-clockwise directions. We observe that if we look at the trajectory on and project it to , we obtain a billiard trajectory, as defined in §2.2, see Figure 2. We refer the reader to [1] and [2] for more details.

As we will see in §2.2, we can define a toric action on the set of points belonging to a billiard trajectory in . Given such a point which is not on the two trajectories contained in , we would like to define two circle actions as follows. The first one is given by rotating and the corresponding by the same angle. The other one is given by moving along the billiard trajectory and rotating back by an angle whose proportion to the total angle spanned by the line segment is equal to the amount moved on it. These two actions correspond to translations in the toric coordinates and , respectively, which will be defined in §2.2.

132
Figure 2: The segments representing a billiard trajectory

1.3 Ball packings and ECH capacities

ECH capacities of a concave toric domain can be computed using an appropiate ball packing, as explained in [4]. We now recall this construction and compute the first two ECH capacities of .

Let be a concave toric domain. The weight expansion of is the multiset defined as follows. For a triangle with vertices , and , we define . If , we let . Let be the largest triangle contained in . Then , where and could be empty. We translate the closures of and so that the obtuse corners are at the origin and we multiply these regions by the matrices and , respectively, obtaining two regions that we call and , respectively. In particular, and are also concave toric domains. Assuming that and are defined, we let

Here we consider the union with repetition. We now proceed by induction to define and in terms of triangles and smaller regions. This process is infinite, unless is bounded by the graph of a piecewise linear function whose slopes are all rational.

We now define the weight sequence to be the non-increasing ordering of the elements of . As explained in [4] and reviewed in §3, for every , there exists a symplectic embedding

(3)

Therefore for every ,

(4)
Remark 6.

Since is a non-increasing sequence, it follows from (1) that

Therefore the limit in (4) equals .

The main theorem of [4] is the following.

Theorem 7 (Choi, Cristofaro-Gardiner, Frenkel, Hutchings, Ramos [4]).

Let be a concave toric domain and let be the weight sequence of . Then for every ,

(5)

In §3, we will show that the first two weights of are:

(6)

From (5) and (6) it follows that:

(7)

We note that (7) is enough to obtain the ’only if’ parts of Theorem 1 as we explain below.

1.4 Proof of Theorem 1

The last ingredient of the proof of Theorem 1 is the following proposition.

Proposition 8.

There exists a symplectic embedding .

Assuming Theorem 3 and Proposition 8, we can now prove Theorem 1.

Proof of Theorem 1.

By Theorem 3, we can substitute in Theorem 1 by the concave toric domain .

(a) First let us assume that symplectically embeds into . Then

Conversely, , for all . So, by Proposition 8, symplectically embeds into for all .

(b) Assume that , where . We recall that and . Hence

The converse is a direct consequence of Proposition 8.

(c) Assume that , where . Again we have

For the converse, we can construct an explicit embedding by

1.5 A converse question

We may also ask a converse question, namely, when ellipsoids embed into the lagrangian bidisk. Although this question is still open in general, we can answer it in two cases.

Corollary 9.

Let . Then ECH capacities give a sharp obstruction to symplectically embedding into . In particular if, and only if, .

Proof.

We assume that . We consider first the case . Then it follows from (7) that . So

The second inclusion above follows from a simple calculation, see Figure 3(a). Now suppose that . From (7) we obtain . So . We also observe that the line is tangent to the curve (2) and hence . Therefore . ∎

Remark 10.

Corollary 9 does not hold for . In fact, one can construct better embeddings than the inclusion by symplectic folding, see [13]. We do not know whether there are better embeddings than the inclusion for .

Remark 11.

Gutt and Hutchings have recently announced a result that implies that if, and only if, .

1.6 Outline of the paper

The rest of this paper is organized as follows. In §2, we prove that is symplectomorphic to the interior of a concave toric domain, namely , thus proving Theorem 3. In §3, we prove Proposition 8, that is, we show that embeds into . As explained in §1.4, this concludes the proof of Theorem 1.

Acknowledgments.

I would like to thank Felix Schlenk for asking the question that inspired this paper and Michael Hutchings for helpful discussions. During the course of this work, I was supported by the European Research Council Grant Geodycon and a grant of the French region Pays de la Loire. I would also like to thank the anonymous referees for very helpful comments and suggestions.

2 Concave toric domains and the lagrangian bidisk

2.1 Outline

In this section, we will prove Theorem 3. We recall that is the product of two lagrangian disks and is the concave toric domain where is the region bounded by the coordinate axes and the curve (2). We will prove that and are symplectomorphic.

The idea is to exhaust by domains which are endowed with a Hamiltonian toric action whose moment image converges to . In other words, for each , we will construct symplectic manifolds and toric domains such that for each , the domains and are symplectomorphic and

The definition of is relatively simple and uses an idea from [3], which was also used in [1]. Let be a smooth function such that:

  • ,

  • , for and for all ,

  • as .

To simplify the notation, we will denote a point in by , where and although the orientation of is still given by . For , let and

We observe that is a Liouville domain with Liouville form

We also note that for , we have , and that .

The definition of is more complicated and it will be given in §2.4. The idea is as follows. We first define a function , for some such that is a Hopf link. Then we define toric coordinates in the complement of this Hopf link. Finally we show that and extend to the different components of this link and that there exist functions and defined on such that

We define to be the toric domain where is the region bounded by the coordinate axes and the image of .

2.2 Billiards

We will now give an idea of how to define the toric coordinates on by taking the limit . We will explain the heuristics in this subsection and give the actual definition that we will use to prove Theorem 3 in §2.3.

The Liouville form is a contact form on each and its Reeb flow is parallel to the Hamiltonian vector field

If is a trajectory of this flow, then . So is determined by the curve which satisfies the equation . As explained in [3] and in [1], a sequence of solutions to this equation with and bounded energy admits a subsequence which converges to a closed billiard trajectory in a suitable topology.

To get an idea of what is happening, we observe that for very small , the acceleration is very close to , except in a neighborhood of . So is very close to a line segment away from and it bends sharply near . At a point of maximum of , we have . It follows from the proof of Lemma 13 below that is symmetric with respect to the reflection about the line spanned by . Moreover

So if we take a family of curves with constant such that the curve for converges to a line segment of direction , then the limiting curve for will also be a line segment and

So is what we call a billiard trajectory.

A billiard trajectory is a curve in which is piecewise smooth and satisfies:

  • and whenever is smooth at .

  • If is not smooth at , then and

Let be the space of points that belong to a billiard trajectory of . Here is the velocity of the billiard trajectory at . A natural pair of commuting independent Hamiltonians for the billiard flow on the disk is where and is the angular momentum. But the induced action is not toric. In fact the vector field induces an -action which is usually not periodic. We can use to produce a pair of Hamiltonians which generate a toric action. We do that indirectly by defining explicit action-angle coordinates as we explain below.

We let . Let be the set of the points in corresponding to the oriented line segment from to . For , we define to be the ratio . It follows from a simple calculation that

We also define .

We can see as a subset of . Under this inclusion, it follows from a calculation using the definitions above that

In order to obtain an actual toric domain, we need to perform a change of variables:

So

(8)

In the following sections, we will make these ideas precise and explain how an equation such as (8) implies that is symplectomorphic to a toric domain.

2.3 The toric coordinates

We now fix and we let endowed with the contact form

For , we define , where denotes the two-dimensional cross-product. We observe that is constant along the Reeb trajectories.

Lemma 12.

The function takes values in for some . Moreover and are circles.

Proof.

The Reeb flow is parallel to the vector field

An integral curve of is a solution to the system of differential equations:

(9)

In particular, a solution to (9) is determined by its projection , which satisfies

(10)

Moreover, when specifying the initial conditions to (10), it is enough to give the direction of since its length is determined by the fact that .

Let be a parametrization of an integral curve of . We observe that

(11)
(12)

It follows from (11) and (12) that always has a maximum. We note that the points of local extrema of are the same as the ones of , but if , then is not smooth for such that .

If is a point of maximum or minimum of , then for every ,

where and is the sign of . For , let . It follows from our choice of that is an odd function and that it has exactly two critical points and . Moreover and are the points of global maximum and minimum, respectively. Let . So takes values in .

Let . We will show that and are circles. Let be the integral trajectory of such that and let be a point of maximum of . So . From it follows that

So , where denotes complex multiplication in the plane . Now let

(13)

Then satisfies (10) and . By the uniqueness of solutions of differential equations, . So is a circle. Analogously, we can show that is a circle.

Let be the circles defined above and let . We will show that is a torus bundle and we will define a trivialization . In other words, we will construct a diffeomorphism . Before doing that, we will prove a lemma that will be necessary for the definition of the functions and .

Lemma 13.

Let be a Reeb trajectory.

  1. If are two consecutive points of maximum of , then the differences and are independent of the choice of the pair .

  2. The differences in (a) depend only on the value of .

Proof.

(a) Let be a Reeb trajectory and let be a parametrization of the same curve, but now as an integral curve of , i.e.,

for some smooth function . By a simple computation, we obtain:

where . So

(14)

We write in polar coordinates . It follows from (14) that (10) is equivalent to the following system of equations:

(15)

Now let be three consecutive points of maximum of . By a translation of time, we can assume without loss of generality that . We now let and . We observe that satisfies (15). Moreover

By the uniqueness of solutions of differential equations, we conclude that and . So

(16)

Now, since and and since there are no points of maximum of in , it follows that is a point of maximum of and that there are no other points of maximum in the interval . So and . By induction, we conclude that the difference between any two consecutive points of maximum of is always and that the difference between their -values is .

(b) We first claim that if are such that , then is on the same Reeb trajectory as for some . Let and be Reeb trajectories going through and at time , respectively. We can assume without loss of generality that is a point of minimum of and . We assumed that . It follows from (11) that for . For , we recall that . Since , we have

(17)

where is the sign of . We observe that