A symplectic embedding of the cube with minimal sections and a question by Schlenk

I prove that the open unit cube can be symplectically embedded into a longer polydisc in such a way that the area of each section satisfies a sharp bound and the complement of each section is path-connected. This answers a variant of a question by F. Schlenk.


The main result
Let n ≥ 2. By q 1 , p 1 , . . . , q n , p n we denote the standard coordinates in R 2n , and we equip R 2n with the standard symplectic form ω 0 := n i=1 dq i ∧ dp i . 1 We denote by B m r resp. B m r the open resp. closed ball in R m of radius r around 0. M. Gromov's famous non-squeezing theorem [2,Corollary,p. 310] implies that B 2n r does not symplectically embed into the closed unit symplectic cylinder B 2 1 ×R 2n−2 if r > 1. In [5] F. Schlenk investigated how flexible symplectic embeddings are in the case r ≤ 1. More precisely, for every z ∈ R 2n−2 , we define ι z : R 2 → R 2n , ι z (y) := (y, z).
Answering a question of D. McDuff [4], in [5, Theorem 1.1] Schlenk proved that for every a > 0, there exists a symplectic embedding ϕ of B This article is part of the topical collection "Symplectic geometry-A Festschrift in honour of Claude Viterbo's 60th birthday" edited by Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk. 1 Following the physicists' convention, I use an upper index for the i-th coordinate of a point q in the base manifold R n and lower index for the i-th coordinate of a covector p ∈ R n = T * q R n . 2 This means two-dimensional Lebesgue measure. The main result of this article is the following. This theorem answers Questions 1 and 2 affirmatively. It also provides a negative answer to Schlenk's Question 3 with the word "closure" dropped. It even implies that there exists a symplectic embedding for which the bounded hull of each section has arbitrarily small area: (For a proof see p. 9.) This corollary is optimal in the sense that its statement becomes false if we replace B 2n 1 and B 2 1 by the closed balls B contains the circle of radius 1 around 0. 3 There is always a section of area at least 1 c , by Fubini's theorem. Hence, a = 1 c is the minimal possible bound. 4 We do not impose any restrictions on how ϕ maps the boundary of the ball. In particular, the bounded hull of this section equals B 2 1 , which has area π.
Remark. Let ϕ be as in the statement of Theorem 4. Then each section of the image of ϕ equals its own bounded hull. Hence, ϕ is a sharp counterexample to a variant of Question 3 concerning embeddings of cubes.
In the case n = 2, the idea of proof of Theorem 4 is to consider the linear symplectic map Ψ : (q, p) → (Q, P ) induced by the Lagrangian shear p → P := p 1 , cp 1 + p 2 . The P 2 -sections of the image of the square (0, 1) 2 under this shear have length at most 1 c . Hence, the area of each section of Ψ (0, 1) 4 is at most 1 c . To make the image of Ψ fit in the polydisc (0, 1) 3 × (0, c), we wrap its upper part (in P 2 -direction) back to the lower part, by passing to the quotient R/cZ. We also wrap the Q 1 -coordinate. See Fig. 1.
Finally, we compose the resulting map with the product of two area preserving embeddings of finite cylinders into rectangles. This yields a symplectic embedding with the desired properties.
• This construction is similar to L. Traynor's symplectic wrapping construction, which she used e.g. to show that certain polydiscs embed into certain cubes, see [7] and [6,Chap. 7]. One difference is that I wrap coordinates of mixed type (Q and P ), whereas Traynor wraps coordinates of pure type. • Schlenk proved a nonsharp result regarding the areas of the bounded hulls of the sections. More precisely, his folding method [6, Sect. 8.3] can be used to prove that for every n ≥ 2, positive integer k, and ∈ (0, 1) there exists a symplectic embedding ϕ : (0, ) 2n → (0, 1) 2n−1 × (0, k), such that the bounded hull of every section of ϕ (0, ) 2n has area at most 1 k . Theorem 4 improves this in the following ways: -It treats the critical case = 1.
-It makes the area estimate sharp.
-It holds for any real number c ≥ 1, not only for an integer c = k.
-The proof of Theorem 4 is easier than the folding method.
• In [5] and [6, p. 226], Schlenk calls the bounded hull of the closure of a set its "simply connected hull". The simply connected hull of a simply connected compact subset S of R m need not be equal to S. In the case m ≥ 3, an example is given by the sphere S := S m−1 , and in the case m = 2 by the Warsaw circle. This set is produced by closing up the topologist's sine curve with an arc. For this reason, I prefer the terminology "bounded hull". Since this notion is only defined for bounded subsets of R m , no confusion should arise from the fact that the bounded hull of a bounded set S can differ from S. • For more information about related work, see [6].

Proofs of Theorem 4 and of Corollary 5
In the proofs of Theorem 4 and Corollary 5, we will use the following lemma. The idea of proof of this lemma is explained by Fig. 2.
In the proof of Lemma 7, we will use the following. To see this, observe that the map is a homeomorphism that restricts to a diffeomorphism from [0, The desired map θ can be constructed from four copies of θ (one for each corner), using charts for B 2 r and a cut off argument.
Proof of Lemma 7. To prove (i), we define r := π − 1 2 and choose a map θ as in Remark 8. We define Hence, the hypotheses of Proposition 9 are satisfied. We choose a diffeomorphism ϕ as in the statement of this proposition. The map has the required properties.
We prove (ii). There exists a symplectomorphism We choose a symplectomorphism ξ of [0, 1] 2 that equals the identity in a neighbourhood of the boundary and maps κ(0) to y 0 . We obtain such a map as the Hamiltonian flow of a suitable function on (0, 1) 2 with compact support. The map has the required properties. This proves (ii) and completes the proof of Lemma 7.
Remark. The proof of part (i) of Lemma 7 is based on Proposition 9. The proof of that result in turn uses Moser isotopy and a lemma that roughly states that a primitive of an exact top degree form can be chosen in such a way that it vanishes on the boundary of the manifold. An alternative approach for proving Lemma 7(i) is based on the proof of [6, Lemma 3.1.5]. That lemma states the following. We define U := B 2 r and V := (0, 1) 2 . The idea of the alternative approach to Lemma 7(i) is to choose admissible families of loops in such a way that the symplectomorphism constructed in the proof of Lemma 10 extends continuously and injectively to the closure of U . (Neither condition is automatically satisfied.) The extension will then have the desired properties.
Proof of Theorem 4. Consider first the case n = 2. We denote by π : the canonical projection, and equip (R/Z) × R × R × (R/cZ) with the symplectic form induced by ω 0 and π. We denote y 0 := z 0 := 1 2 , 1 2 . We choose a map λ as in Lemma 7(ii). It follows from the same lemma that there exists a symplectomorphism The map ϕ is well defined, since π • Ψ maps (0, 1) 4 to the product of the domains of λ and λ . The map ϕ is a symplectic immersion, as it is the composition of three symplectic immersions. A straight-forward argument shows that π •Ψ (0, 1) 4 is injective. Since λ|(R/Z)×(0, 1) and λ are injective, it follows that the same holds for ϕ. Hence, ϕ is a symplectic embedding of (0, 1) 4 into (0, 1) 3 × (0, c).