A symplectic embedding of the cube with minimal sections and a question by Schlenk
aa r X i v : . [ m a t h . S G ] M a y A SYMPLECTIC EMBEDDING OF THE CUBE WITHMINIMAL SECTIONS AND A QUESTION BYSCHLENK
FABIAN ZILTENER
Abstract.
I prove that the open unit cube can be symplecticallyembedded into a longer polydisc in such a way that the area ofeach section satisfies a sharp bound and the complement of eachsection is path-connected. This answers a variant of a question byF. Schlenk. The main result
Let n ≥
2. By q , p , . . . , q n , p n we denote the standard coordinatesin R n , and we equip R n with the standard symplectic form ω := P ni =1 dq i ∧ dp i . We denote by B mr resp. B mr the open resp. closed ball in R m of radius r around 0. M. Gromov’s famous nonsqueezing theorem[Gro85, Corollary, p. 310] implies that B nr does not symplecticallyembed into the closed unit symplectic cylinder B × R n − if r > r ≤
1. More precisely, for every z ∈ R n − we define ι z : R → R n , ι z ( y ) := ( y, z ) . Answering a question by D. McDuff [McD98], in [Sch03, Theorem 1.1]Schlenk proved that for every a > ϕ of B n into B × R n − , such that for every z ∈ R n − the section ι − z (cid:0) ϕ (cid:0) B n (cid:1)(cid:1) has area at most a .Schlenk’s lifting method [Sch05, Section 8.4] also shows that for everypositive integer k and every a > k there exists a symplectic embeddingof the open cube (0 , n into the open polydisc (0 , n − × (0 , k ), whosesections have area at most a . The main result of the present articleanswers the following two questions: Question 1.
Is this statement true with the integer k replaced by ageneral real number c ≥ ? Following the physicists’ convention I use an upper index for the i -th coordinateof a point q in the base manifold R n and lower index for the i -th coordinate of acovector p ∈ R n = T ∗ q R n . This means two-dimensional Lebesgue measure.
Question 2.
Can the bound a on the areas of the sections be madesharp, i.e., equal to c ? I also answer a variant of the following question by Schlenk. Forevery bounded subset S of R m we define the bounded hull of S to bethe union of S and all bounded connected components of R m \ S . Question 3 (Schlenk, [Sch03], Question 2.2) . Let n ≥ , ϕ be a sym-plectic embedding of B n into B × R n − , and a < π . Does thereexist z ∈ R n − such that the bounded hull of the closure of the section ι − z (cid:0) ϕ (cid:0) B n (cid:1)(cid:1) has area at least a ? The main result of this article is the following.
Theorem 4.
For every n ≥ and c ∈ [1 , ∞ ) there exists a symplecticembedding ϕ : (0 , n → (0 , n − × (0 , c ) , such that for every z ∈ R n − the following holds:(i) The section ι − z (cid:0) ϕ (cid:0) (0 , n (cid:1)(cid:1) has area at most c .(ii) Its complement in R is path-connected. This theorem answers Questions 1 and 2 affirmatively. It also pro-vides a negative answer to Schlenk’s Question 3 with the word “closure”dropped. It even implies that there exists a symplectic embedding forwhich the bounded hull of each section has arbitrarily small area:
Corollary 5.
For every n ≥ and a > there exists a symplecticembedding ψ : B n → B × R n − , such that the bounded hull of eachsection of ψ (cid:0) B n (cid:1) has area at most a . (For a proof see p. 8.) This corollary is optimal in the sense that itsstatement becomes false if we replace B n and B by the closed balls B n and B . Even the following is true: Proposition 6 (F. Lalonde, D. McDuff) . Let n ∈ N and ϕ : B n → B × R n − be a symplectic embedding. Then there exists z ∈ R n − ,such that the section ι − z (cid:0) ϕ (cid:0) B n (cid:1)(cid:1) contains the circle of radius around0. In particular the bounded hull of this section equals B , which hasarea π . Proof of Proposition 6.
This follows from [LM95, Lemma 1.2]. (cid:3)
Remark.
Let ϕ be as in the statement of Theorem 4. Then each sectionof the image of ϕ equals its own bounded hull. Hence ϕ is a sharp counterexample to a variant of Question 3 concerning embeddings ofcubes. There is always a section of area at least c , by Fubini’s theorem. Hence a = c is the minimal possible bound. We don’t impose any restrictions on how ϕ maps the boundary of the ball. SYMPLECTIC EMBEDDING OF THE CUBE WITH MINIMAL SECTIONS 3 } Figure 1.
The green arrow depicts the Lagrangianshear p P := (cid:0) p , cp + p (cid:1) , and the orange arrowthe induced shear in the q -plane. The black arrows de-pict the wrapping maps. The magenta line segment is a P -section of the image of the square under the composedmap in the p -plane, where P ∈ R / ( c Z ). The violet setdepicts a Q -section of the image of the open square un-der the composed map in the q -plane, where Q ∈ R / Z .The bracket } indicates that the product of these twosets is given by the red ribbon on the blue cylinder. Theimage of this ribbon under some area-preserving map isa section the image of the desired symplectic embedding ϕ . It has area equal to c .In the case n = 2 the idea of proof of Theorem 4 is to consider thelinear symplectic map Ψ : ( q, p ) ( Q, P ), induced by the Lagrangianshear p P := (cid:0) p , cp + p (cid:1) . The P -sections of the image of thesquare (0 , under this shear have length at most c . Hence the areaof each section of Ψ (cid:0) (0 , (cid:1) is at most c . To make the image of Ψ fitin the polydisc (0 , × (0 , c ), we wrap its upper part (in P -direction)back to the lower part, by passing to the quotient R /c Z . We also wrapthe Q -coordinate. See Figure 1. Finally, we compose the resulting mapwith the product of two area preserving embeddings of finite cylindersinto rectangles. This yields a symplectic embedding with the desiredproperties. FABIAN ZILTENER
Remarks (method of proof, related result, terminology) . • Thisconstruction is similar to L. Traynor’s symplectic wrapping con-struction, which she used e.g. to show that certain polydiscs em-bed into certain cubes, see [Tra95] and [Sch05, Chapter 7] . Onedifference 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 thebounded hulls of the sections. More precisely, his folding method [Sch05, Section 8.3] can be used to prove that for every n ≥ ,positive integer k , and ℓ ∈ (0 , there exists a symplectic em-bedding ϕ : (0 , ℓ ) n → (0 , n − × (0 , k ) , such that the boundedhull of every section of ϕ (cid:0) (0 , ℓ ) n (cid:1) has area at most k . Theorem4 improves this in the following ways: – It treats the critical case ℓ = 1 . – It makes the area estimate sharp. – It holds for any real number c ≥ , not only for an integer c = k . – The proof of Theorem 4 is easier than the folding method. • In [Sch03] and [Sch05, p. 226] Schlenk calls the bounded hullof the closure of a set its “simply connected hull”. The simplyconnected hull of a simply connected compact subset S of R m need not be equal to S . In the case m ≥ an example is givenby the sphere S := S m − , and in the case m = 2 by the Warsawcircle. This set is produced by closing up the topologist’s sinecurve 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 thebounded hull of a bounded set S can differ from S . • For more information about related work see [Sch05] . Proofs of the main result and of Corollary 5
In the proofs of Theorem 4 and Corollary 5 we will use the followinglemma.
Lemma 7 (squaring the disc and the cylinder) . We denote r := π − .(i) There exists a homeomorphism κ : B r → [0 , , that restricts to a (smooth) symplectomorphism between the inte-riors.(ii) For every y ∈ (0 , there exists continuous map λ : ( R / Z ) × [0 , → [0 , that maps ( R / Z ) × { } to y , and restricts to a homeomorphismfrom ( R / Z ) × [0 , to [0 , \ { y } and to a symplectomorphismfrom ( R / Z ) × (0 , to (0 , \ { y } . SYMPLECTIC EMBEDDING OF THE CUBE WITH MINIMAL SECTIONS 5
Figure 2.
The two arrows depict area-preservingsmooth embeddings whose composition is an area-preserving embedding of the open cylinder into the opensquare. The idea of proof of Lemma 7 is to choose suchmaps in such a way that they continuously extend to theclosed cylinder and the closed disc, respectively.The idea of proof of this lemma is explained by Figure 2. In theproof of Lemma 7 we will use the following.
Remark 8 (straightening corners) . We denote by Σ the square [0 , without the corners. Let r > and S be a subset of the circle of radius r consisting of four points. There exists homeomorphism θ : [0 , → B r that restricts to a diffeomorphism from Σ onto B r \ S , such that ( θ | Σ) ∗ ω extends to a nonvanishing smooth 2-form on B r .To see this, observe that the map e θ : [0 , ∞ ) → R × [0 , ∞ ) , e θ ( z ) := z | z | , is a homeomorphism that restricts to a diffeomorphism from [0 , ∞ ) \{ } onto (cid:0) R × [0 , ∞ ) (cid:1) \ { } , that satisfies (cid:0)e θ (cid:12)(cid:12) [0 , ∞ ) \ { } (cid:1) ∗ ω = ω . The desired map θ can be constructed from four copies of e θ (one foreach corner), using charts for B r and a cut off argument. Proposition 9 (Banyaga’s Moser stability with boundary) . Let M be a compact connected oriented smooth manifold, and Ω , Ω volumeforms on M satisfying Z M Ω = Z M Ω . Then there exists diffeomorphism ϕ of M satisfying ϕ ∗ Ω = Ω , ϕ | ∂M = id . Proof.
See [Ban74, Th´eor`eme, p. 127]. (cid:3)
FABIAN ZILTENER
Proof of Lemma 7.
To prove (i) , we define r := π − and choose a map θ as in Remark 8. We define M := B r , Ω := θ ∗ ω , Ω := ω . We have Z M Ω = Z Σ ω = 1 = Z M Ω . Hence the hypotheses of Proposition 9 are satisfied. We choose a dif-feomorphism ϕ as in the statement of this proposition. The map κ := ( ϕ ◦ θ ) − : B r → [0 , has the required properties.We prove (ii) . There exists a symplectomorphism χ : ( R / Z ) × [0 , → B r \ { } . For example, consider y : R / Z → C = R , y (¯ q ) := e πiq , where q ∈ ¯ q isan arbitrary representative, and define χ ( q, p ) := r p − py ( q ) . We choose a symplectomorphism ξ of [0 , that equals the identityin a neighbourhood of the boundary and maps κ (0) to y . We obtainsuch a map as the Hamiltonian flow of a suitable function on (0 , with compact support. The map λ := (cid:26) ξ ◦ κ ◦ χ on ( R / Z ) × [0 , ,y on ( R / Z ) × { } has the required properties. This proves (ii) and completes the proofof Lemma 7. (cid:3) Proof of Theorem 4.
Consider the case n = 2. We denote by π : R → ( R / Z ) × R × R × ( R /c Z )the canonical projection, and equip ( R / Z ) × R × R × ( R /c Z ) with thesymplectic form induced by ω and π . We denote y := z := (cid:0) , (cid:1) .We choose a map λ as in Lemma 7(ii). It follows from the same lemmathat there exists a symplectomorphism λ ′ : (0 , × ( R /c Z ) → (cid:0) (0 , × (0 , c ) (cid:1) \ { z } . We defineΨ : R → R , Ψ (cid:0) q , p , q , p (cid:1) := (cid:0) q − cq , p , q , cp + p (cid:1) ,ϕ := ( λ × λ ′ ) ◦ π ◦ Ψ (cid:12)(cid:12) (0 , . The map ϕ is well-defined, since π ◦ Ψ maps (0 , to the product ofthe domains of λ and λ ′ . The map ϕ is a symplectic immersion, as itis the composition of three symplectic immersions. A straight-forwardargument shows that π ◦ Ψ (cid:12)(cid:12) (0 , is injective. Since λ | ( R / Z ) × (0 , ξ is a smooth map in the sense of manifolds with boundary and corners. SYMPLECTIC EMBEDDING OF THE CUBE WITH MINIMAL SECTIONS 7
Figure 3.
The arrow depicts the area preserving map λ : ( R / Z ) × (0 , → (0 , . (Compare to Figure 2.) Itsends the upper part of the cylinder close to the centerof the disc. The red ribbon on the cylinder is U Q ,P ,the section of the image of ϕ . The point P determinesthe height of the upper boundary of the red ribbon, andtherefore the radius of the circle inside the square. Thepoint Q determines the position of the blue slit. Becauseof this slit, the blue set on the right is path-connected.This is the complement of the image of the section underthe map λ .and λ ′ are injective, it follows that the same holds for ϕ . Hence ϕ is asymplectic embedding of (0 , into (0 , × (0 , c ).Let ( Q , P ) ∈ (0 , × ( R /c Z ). We have U Q ,P := (cid:8) ( Q , P ) ∈ ( R / Z ) × (0 , (cid:12)(cid:12) (cid:0) Q , P , Q , P (cid:1) ∈ π ◦ Ψ (cid:0) (0 , (cid:1)(cid:9) = V Q × W P , (1)(2) V Q := (cid:8) q − cQ + Z (cid:12)(cid:12) q ∈ (0 , (cid:9) = ( R / Z ) \ {− cQ + Z } ,W P := (cid:8) P ∈ (0 , (cid:12)(cid:12) ∃ p ∈ (0 ,
1) : cP + p + c Z = P (cid:9) (3) = (0 , ∩ [ p ∈ (0 , P − p c , where P − p c ∈ R / Z . The set W P is an open subinterval of (0 ,
1) or theunion of two such subintervals. It has length c . Using (1) and (2), itfollows that U Q ,P has area equal to c . Figure 3 depicts the set U Q ,P .Let now z ∈ (cid:0) (0 , × (0 , c ) (cid:1) \ { z } . We denote ( Q , P ) := λ ′− ( z ).We have(4) λ − (cid:0) ι − z (cid:0) ϕ (cid:0) (0 , (cid:1)(cid:1)(cid:1) = U Q ,P . FABIAN ZILTENER
Since λ is area-preserving, it follows that the section ι − z (cid:0) ϕ (cid:0) (0 , (cid:1)(cid:1) has area equal to c . For z = z or z outside of (0 , × (0 , c ), thesection is empty. This proves (i).To prove property (ii), consider the continuous path y : [0 , → [0 , , y ( t ) := λ (cid:0) − cQ + Z , t (cid:1) . The point y (0) lies on the boundary of the square [0 , . It followsfrom (2) that the path y lies inside the complement of ι − z (cid:0) ϕ (cid:0) (0 , (cid:1)(cid:1) in R . Every point outside (0 , can be connected to y (0) througha continuous path outside of (0 , . Every point in the complementof ι − z (cid:0) ϕ (cid:0) (0 , (cid:1)(cid:1) in (0 , can be connected to a point on the path y through a path in this complement. This follows from (4) and the facts U Q ,P = V Q × W P , V Q = ( R / Z ) \ {− cQ + Z } . See again Figure 3.This proves (ii).Hence ϕ has the desired properties. This proves Theorem 4 in thecase n = 2. For n ≥ ϕ with the identitymap. (cid:3) In the proof of Corollary 5 we will use the following.
Remark 10 (monotonicity) . The bounded hull is monotone in thesense that if A ⊆ B ⊆ R m are bounded sets then the bounded hullof A is contained in the bounded hull of B .Proof of Corollary 5. We define r := π − . By a rescaling argumentit suffices to show that for every a ∈ (0 ,
1] there exists a symplecticembedding ψ : B nr → B r × R n − , such that the bounded hull of eachsection of ψ ( B nr ) has area at most a . To prove this statement, wechoose ϕ is as in the conclusion of Theorem 4 with c := a . We choosea map κ as in Lemma 7(i). The map ψ := ( κ − × id) ◦ ϕ ◦ (cid:0) κ × · · · × κ (cid:1) : B nr → B r × R n − is a symplectic embedding. Let z ∈ R n − . Property (ii) in Theo-rem 4 implies that the complement of V := κ − (cid:16) ι − z (cid:0) ϕ (cid:0) (0 , n (cid:1)(cid:1)(cid:17) in R is path-connected. Hence V equals its bounded hull. The section ι − z (cid:0) ψ ( B nr ) (cid:1) is contained in V . Using Remark 10, it follows that thebounded hull of this section is also contained in V . Using Theorem4(i) and that κ is area-preserving, it follows that this bounded hull hasarea at most c = a . Hence ψ has the desired properties. This provesCorollary 5. (cid:3) Acknowledgments
I would like to thank Felix Schlenk for an interesting discussion andfor proof-reading the first version of this article.
SYMPLECTIC EMBEDDING OF THE CUBE WITH MINIMAL SECTIONS 9
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