Symplectic embeddings of -dimensional ellipsoids into cubes
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
- 1 Introduction
- 2 Proof of Proposition 1.4 and equalities (1.1)
- 3 Reduction to a constraint function given by exceptional spheres
- 4 Basic observations
- 5 The interval
- 6 The interval
- 7 The interval
- A Computer programs
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
(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
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.
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 .
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.
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.
Let with rational. Then there exists a symplectic embedding if and only if there exists a symplectic embedding
There exists a symplectic embedding if and only if for all .
As a further corollary we obtain
The ellipsoid symplectically embeds into the cube if and only if symplectically embeds into the ellipsoid .
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
For the easy proof, we refer to Section 2.∎
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.
2.1. Decomposing an ellipsoid into a disjoint union of balls
In [M2], McDuff showed the following theorem.
(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].
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 .
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
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.
(McDuff-Schlenk [MS], Lemma 1.2.6) Let be a rational number with relatively prime, and let be its weight expansion. Then
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
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
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 .
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 .
“ 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.
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
2.4. Proof of equalities (1.1)
For all ,
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.
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 .
(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 .
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 .
In order to give a characterization of the set , we need the following definition as in [MS].
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 .
(McDuff-Schlenk [MS], Proposition 1.2.12 and Remark 3.3.1)
All satisfy the two Diophantine equations
For all distinct we have
A tuple belongs to if and only if satisfies the Diophantine equations in (i) and can be reduced to by repeated Cremona moves.
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