Symplectic embeddings of 4-dimensional ellipsoids into cubes

Symplectic embeddings of -dimensional ellipsoids into cubes

David Frenkel and Dorothee Müller
Abstract.

Recently, McDuff and Schlenk determined in [MS] the function whose value at is the infimum of the size of a -ball into which the ellipsoid symplectically embeds (here, is the ratio of the area of the large axis to that of the smaller axis of the ellipsoid). In this paper we look at embeddings into four-dimensional cubes instead, and determine the function whose value at is the infimum of the size of a -cube into which the ellipsoid symplectically embeds (where denotes the disc in of area ). As in the case of embeddings into balls, the structure of the graph of is very rich: for less than the square  of the silver ratio , the function turns out to be piecewise linear, with an infinite staircase converging to . This staircase is determined by Pell numbers. On the interval , the function coincides with the volume constraint except on seven disjoint intervals, where is piecewise linear. Finally, for , the functions and are equal.

For the proof, we first translate the embedding problem to a certain ball packing problem of the ball . This embedding problem is then solved by adapting the method from [MS], which finds all exceptional spheres in blow-ups of the complex projective plane that provide an embedding obstruction.

We also prove that the ellipsoid symplectically embeds into the cube if and only if symplectically embeds into the elllipsoid . Our embedding function thus also describes the smallest dilate of into which symplectically embeds.

Key words and phrases:
Symplectic embeddings, Pell numbers
2000 Mathematics Subject Classification:
53D05, 14B05, 32S05, 11A55
Partially supported by SNF grant 200020-132000.

1. Introduction

1.1. Statement of the result

Let be the Euclidean -dimensional space endowed with the canonical symplectic form . Any open subset of is also endowed with . Simple examples are the symplectic cylinders (where is the open disc of area ), the open symplectic ellipsoids

and the open polydiscs . We denote the open ball (of radius ) by and the open cube by . Since is symplectomorphic to an open square, is indeed symplectomorphic to a cube.

Given two open subsets and , we say that a smooth embedding is a symplectic embedding if preserves , that is, if . In the sequel, we will write for such an embedding. Since symplectic embeddings are volume preserving, a necessary condition for the existence of a symplectic embedding is, of course, , where . For volume preserving embeddings, this is the only condition (see e.g. [S1]). For symplectic embeddings, however, the situation is very different, as was detected by Gromov in [G]. Among many other things, he proved

Example 1.1.

(Gromov’s nonsqueezing Theorem) There exists a symplectic embedding of the ball into the cylinder if and only if .

Notice that the volume of the cylinder is infinite, and that for any the ball embeds by a linear volume preserving embedding into . Similarly, we also have

Example 1.2.

There exists a symplectic embedding of the ball into the cube if and only if .

The above results show that symplectic embeddings are much more special and in some sense “more rigid” than volume preserving embeddings. A next step was to understand this rigidity better. One way of doing this is to fix a domain of finite volume, and to try to determine for each the -th packing number

Here, is the disjoint union of equal balls . It follows from Darboux’s Theorem that always . If , one says that admits a full packing by balls, and if , one says that there is a packing obstruction. Again, it is known that if we would consider volume preserving embeddings instead, then all packing numbers would always be 1.

In imporant work by Gromov [G], McDuff-Polterovich [MP] and Biran [B] all the packing numbers of the -ball and the -cube were determined. The result for is


This result shows that, while there is symplectic rigidity for many small , there is no rigidity at all for large .

In order to better understand these numbers, we look at a problem that interpolates the above problem of packing by equal balls. For , consider the ellipsoid defined above, and look for the smallest cube  into which symplectically embeds. Since if and only if , we can always assume that , and therefore study the embedding capacity function

on the interval . It is clear that is a continuous and nondecreasing function. Since symplectic embeddings are volume preserving and the volumes of and are and respectively, we must have the lower bound

It is not hard to see that . Therefore, whenever . In [M2], McDuff has shown that the converse is also true! Our ellipsoid embedding problem therefore indeed interpolates the problem of packing by equal balls, and we get

First upper estimates for the function were obtained in Chapter 4.4 of [S2] by explicit embeddings of ellipsoids into a cube. These upper estimates also suggested that symplectic rigidity for the problem should disappear for large .

In this paper, we completely determine the function . In order to state our main theorem, we introduce two sequences of integers: the Pell numbers and the half companion Pell numbers , which are defined by the recurrence relations

Thus, , , and , , , . The two sequences and are then defined by

The first terms in these sequences are

More generally, for all ,

and both sequences converge to , which is the square of the silver ration .

Theorem 1.3.
  1. On the interval ,

    for all (see Figure 1.1).

  2. On the interval we have except on seven disjoint intervals, where is piecewise linear (see Figure 1.2).

  3. For we have .

The proof of (i) is given in Corollary 5.2, a more detailed statement as well as the proof of (ii) are given in Theorem 7.2, while the proof of (iii) is given in Lemma 4.1 and Proposition 7.7.

A similar result has been previously obtained by McDuff-Schlenk in [MS] for the embedding problem . These two results show that the structure of symplectic rigidity can be very rich.

Figure 1.1. The graph of on the interval
Figure 1.2. The graph of on the interval

1.2. Relations to ECH-capacities

There is a more combinatorial (but non-explicit) way of describing the embedding function . Indeed, in [H1], Hutchings used his embedded contact homology to construct for each domain a sequence of symplectic capacities , which for the ellipsoid and the polydisc are as follows.

Form the sequence by arranging all numbers of the form with , in nondecreasing order (with multiplicities). Then for , the -th ECH-capacity is the -th entry of . For instance, .

Moreover, for polydiscs,

There exists a canonical way to decompose an ellipsoid with rational into a finite disjoint union of balls with weights  related to the continued fraction expansion of . We shall explain this decomposition in more detail and prove the following proposition in the next section.

Proposition 1.4.

Let with rational. Then there exists a symplectic embedding if and only if there exists a symplectic embedding

Hutchings showed in Corollary 11 of [H2] how Proposition 1.4 implies that ECH-capacities form a complete set of invariants for the problem of symplectically embedding an ellipsoid into a polydisc:

Corollary 1.5.

There exists a symplectic embedding if and only if for all .

It seems to be hard to derive Theorem 1.3 from Corollary 1.5 or vice-versa.

As a further corollary we obtain

Corollary 1.6.

The ellipsoid symplectically embeds into the cube if and only if symplectically embeds into the ellipsoid .

Proof.

By Corollary 1.5, symplectically embeds into if and only if for all . By McDuff’s proof of the Hofer Conjecture [M3], symplectically embeds into if and only if for all . The corollary now follows from the remark on page 8098 in [H2], that says that for all

(1.1)

For the easy proof, we refer to Section 2.∎

Remark 1.7.

Recall that the ECH capacities of and (or ) are

One sees that the sequence is obtained from by some sort of doubling. This is reminiscent to the doubling in the denition of the Pell numbers: The Fibonacci and Pell numbers are defined recursively by

and while the Fibonacci numbers determine the infinite stairs of the function for (with the golden ratio, see [MS]), the Pell numbers determine the infinite stairs of the function for . This reminiscence may, however, be a coincidence. Indeed, for the ellipsoid the sequence

is obtained from by some sort of trippling, but the beginning of the function describing the embedding problem seems not to be given in terms of numbers defined by .

Acknowledgments. We wish to sincerely thank Felix Schlenk for his precious help and encouragement during the whole project and Régis Straubhaar for his help with all the computer issues.

2. Proof of Proposition 1.4 and equalities (1.1)

In Section 2.1, we explain the canonical decomposition of with into a disjoint union of balls. We then prove Proposition 1.4 in Sections 2.2 and 2.3, and in Section 2.4 we prove equalities (1.1).

2.1. Decomposing an ellipsoid into a disjoint union of balls

In [M2], McDuff showed the following theorem.

Theorem 2.1.

(McDuff [M2]) Let be two rational numbers. Then, there exists a finite sequence of rational numbers such that the closed ellipsoid symplectically embeds into the ball if and only if the disjoint union of balls symplectically embed into .

The disjoint union is then denoted by . Following [MS], we will now explain one way to compute the weights in this decomposition. Notice that in [M2], the weights of the balls are defined in a slightly different way. The proof that these weights agree with the weight expansion of defined now can be found in the Appendix of [MS].

Definition 2.2.

Let be a rational number written in lowest terms. The weight expansion of is the finite sequence defined recursively by

  • , and for all ;

  • if (where we set ), then

  • the sequence stops at if the above formula gives .

Remark 2.3.

If we regard this weight expansion as consisting of blocks on which the are constant, that is

then , and if we set , then for all , . Moreover, the lengths of the blocks give the continued fraction of since

Example 2.4.

The weight expansion of is . The continued fraction expansion of is thus . Notice that we also have

This is no accident and is best explained geometrically as in Figure 2.1. The general result is stated in the next lemma.

Figure 2.1. The weight expansion of .
Lemma 2.5.

(McDuff-Schlenk [MS], Lemma 1.2.6) Let be a rational number with relatively prime, and let be its weight expansion. Then

  1. ;

  2. ;

  3. .

2.2. Representations of balls and polydiscs

In the proof of Proposition 1.4, we shall use certain ways of representing open and closed balls and open polydiscs. Recall that is the open ball in  of capacity , and that , where is the open disc in  of area .

2.2.1. Representations as products

Denote by the open square in . Since is symplectomorphic to the open square , the polydisc is symplectomorphic to

Next, consider the simplex

Then is symplectomorphic to the product

see [T] and Remark 9.3.1 of [S2].

2.2.2. Representations by the Delzant polytope

As before, denote by the Study-Fubini form on the complex projective plane , normalized by . We write for . Its affine part is symplectorphic to the open ball . (Indeed, for , the embedding

is symplectic.)

The image of the moment map of the usual -action on is the closed triangle . For , the preimage of is symplectomorphic to . By precomposing the torus action with suitable linear torus automorphisms, one sees that also the closed triangles based at the other two corners of correspond to closed balls in . We refer to [K] for details.

The image of the moment map of the usual -action on  maps the polydisc to the rectangle .

Figure 2.2. Closed balls in and the moment image of

2.3. Proof of Proposition 1.4

Let now with rational. We need to show that

”: By decomposing into balls as before, we find that , (see also [M2]). Fix . Then also . Now represent the open balls and the polydisc as in Section 2.2.1 above. We then read off from Figure 2.3 that

This holds for every . In view of [M1] we then also find a symplectic embedding .

Figure 2.3.

Assume now that . Fix . Then

According to [M1], the space of symplectic embeddings of into  is connected. Any such isotopy extends to an ambient symplectic isotopy of . In view of this and by Section 2.2.2 we can thus assume that the balls and lie in as shown in Figure 2.4.

Figure 2.4. How and lie in

The image of the balls must then lie over the gray shaded closed region. However, since the balls and are closed, the image of cannot touch the upper horizontal or the right vertical boundary of the gray shaded region. Moreover, according to Remark 2.1.E of [MP] we can assume that this image lies in the affine part of , i.e., the image of the balls lies over the gray region deprived from the dark segment, and hence, by Section 2.2.2, in . We may suppose from the start that . Then . We have thus found a symplectic embedding . It is shown in Theorem 1.5 of [M2] that then also . Hence

It now follows again from [M2] that . (To be precise, [M2] considers embeddings of ellipsoids into open balls; however, the same arguments work for embeddings of ellipsoids into polydiscs.)

2.4. Proof of equalities (1.1)

Lemma 2.6.

For all ,

Proof.

We will prove that and are both equal to the unique integer such that

For , this follows from the fact that the number

of pairs of nonnegative integers such that is equal to . This, in turn, can easily be deduced from the identities

On the other hand, we have by definition that

Fix a nonnegative integer . Let be two nonnegative integers such that

Without loss of generality, . Moreover, we can always take such that . Indeed, assume that with . Then for and , we get

Thus also realizes the minimum. Now, if is even, then and we have to show that

The first inequality follows from the minimality of while the second one follows from the fact that . The case odd is treated similarly. ∎

3. Reduction to a constraint function given by exceptional spheres

In this section we explain how the function can be described by the volume constraint and the constraints coming from certain exceptional spheres in blow-ups of . Since the function is continuous, it suffices to determine for each rational . The starting point is the following lemma, which is a special case of Proposition 1.4.

Lemma 3.1.

Let be a rational number with weight expansion and . Then the ellipsoid embeds symplectically into the cube if and only if there is a symplectic embedding

With this lemma, we have converted the problem of embedding an ellipsoid into a cube to the problem of embedding a disjoint union of balls into a ball. In [MP], the problem of embedding disjoint balls into a ball was reduced to the question of understanding the symplectic cone of the -fold blow-up of . Let be the class of a line, let be the homology classes of the exceptional divisors, and denote by their Poincaré duals. Let be the anti-canonical divisor of , and define the corresponding symplectic cone as the set of classes represented by symplectic forms  with first Chern class .

Theorem 3.2.

(McDuff-Polterovich [MP]) The union embeds into the ball or into if and only if .

To understand , we define as in [MS] the following set .

Definition 3.3.

is the set consisting of and of all tuples with and such that the class is represented in by a symplectically embedded sphere of self-intersection .

We will often write instead of if there is no danger of confusion. We then have the following description of .

Proposition 3.4.

(Li-Li [LiLi], Li-Liu [LiLiu])

In order to give a characterization of the set , we need the following definition as in [MS].

Definition 3.5.

A tuple is said to be ordered if the  are in nonincreasing order. The Cremona transform of an ordered tuple is

A Cremona move of a tuple is the composition of the Cremona transform of with any permutation of the new obtained vector .

Proposition 3.6.

(McDuff-Schlenk [MS], Proposition 1.2.12 and Remark 3.3.1)

  1. All satisfy the two Diophantine equations

  2. For all distinct we have

  3. A tuple belongs to if and only if satisfies the Diophantine equations in (i) and can be reduced to by repeated Cremona moves.

Remark 3.7.

Working directly with Lemma 3.1, Theorem 3.2 and Proposition 3.4 we find, as in [MS], that the only constraints for an embedding are and, for each class ,

(3.1)

One can start from here and use Proposition 3.6 to prove Theorem 1.3. The analysis becomes, however, rather awkward, since the unknown appears on both sides of (3.1).

To improve the situation, we shall apply a base change of , and express the elements of in a new basis. Consider the product (whose affine part is a cube), and form the -fold (topological) blow-up . A basis of