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

TL;DR

The paper demonstrates a symplectic embedding of the open unit cube into a polydisc with optimal section areas and connected complements, addressing a question posed by F. Schlenk.

Contribution

It provides a new symplectic embedding construction that achieves minimal section areas and connected complements, solving a specific open problem.

Findings

Successful symplectic embedding with sharp section bounds

Sections have minimal area as per the established bound

Complement of each section remains path-connected

Abstract

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.

Equations

$${\iota }_z:{\mathbb{R}}^2\to {\mathbb{R}}^{2n},{\iota }_z(y):=(y,z).$$

$$\kappa :{\overline{B}}_r^2\to [0,1{]}^2,$$

$$\lambda :(\mathbb{R}/\mathbb{Z})\times [0,1]\to [0,1{]}^2$$

$$\tilde{\theta }:[0,\infty {)}^2\to \mathbb{R}\times [0,\infty ),\tilde{\theta }(z):=\frac{z^2}{|z|},$$

$$\big{(}{\widetilde{\theta}}\big{|}[0,\infty)^{2}\setminus\{0\}\big{)}_{*}\omega_{0}=\frac{\omega_{0}}{2}.$$

$${\int }_M{\Omega }_0={\int }_M{\Omega }_1.$$

$${\phi }_{\ast }{\Omega }_0={\Omega }_1,\phi |\partial M=\mathrm{id}.$$

$$M:={\overline{B}}_r^2,{\Omega }_0:={\theta }_{\ast }{\omega }_0,{\Omega }_1:={\omega }_0.$$

$${\int }_M{\Omega }_0={\int }_{\Sigma }{\omega }_0=1={\int }_M{\Omega }_1.$$

$$\kappa :=(\phi \circ \theta {)}^{-1}:{\overline{B}}_r^2\to [0,1{]}^2$$

$$\chi :(\mathbb{R}/\mathbb{Z})\times [0,1)\to {\overline{B}}_r^2\setminus \{0\}.$$

$$\chi (\overline{q},p):=r\sqrt{1-p}y(\overline{q}).$$

$$\lambda:=\left\{\begin{array}[]{ll}\xi\circ\kappa\circ\chi&\textrm{ on }(\mathbb{R}/\mathbb{Z})\times[0,1),\\ y_{0}&\textrm{ on }(\mathbb{R}/\mathbb{Z})\times\{1\}\end{array}\right.$$

$$\pi :{\mathbb{R}}^4\to (\mathbb{R}/\mathbb{Z})\times \mathbb{R}\times \mathbb{R}\times (\mathbb{R}/c\mathbb{Z})$$

$$\lambda^{\prime}:(0,1)\times(\mathbb{R}/c\mathbb{Z})\to\big{(}(0,1)\times(0,c)\big{)}\setminus\{z_{0}\}.$$

$$\displaystyle\Psi:\mathbb{R}^{4}\to\mathbb{R}^{4},\quad\Psi\big{(}q^{1},p_{1},q^{2},p_{2}\big{)}:=\big{(}q^{1}-cq^{2},p_{1},q^{2},cp_{1}+p_{2}\big{)},$$

$$\displaystyle\varphi:=(\lambda\times\lambda^{\prime})\circ\pi\circ\Psi\big{|}(0,1)^{4}.$$

$$U_{Q^2,{\overline{P}}_2}:=$$

$$=$$

$$V_{Q^{2}}:=\big{\{}q^{1}-cQ^{2}+\mathbb{Z}\,\big{|}\,q^{1}\in(0,1)\big{\}}=(\mathbb{R}/\mathbb{Z})\setminus\{-cQ^{2}+\mathbb{Z}\},$$

$$W_{{\overline{P}}_2}$$

$$=(0,1)\cap \bigcup _{p_2\in (0,1)}\frac{{\overline{P}}_2-p_2}{c},$$

$$\lambda^{-1}\left(\iota_{z}^{-1}\big{(}\varphi\big{(}(0,1)^{4}\big{)}\big{)}\right)=U_{Q^{2},{\overline{P}}_{2}}.$$

$$y:[0,1]\to[0,1]^{2},\quad y(t):=\lambda\big{(}-cQ^{2}+\mathbb{Z},t\big{)}.$$

$$\psi:=(\kappa^{-1}\times{\operatorname{id}})\circ\varphi\circ\big{(}\kappa\times\cdots\times\kappa\big{)}:B^{2n}_{r}\to B^{2}_{r}\times\mathbb{R}^{2n-2}$$

Full text

1. The main result

Let $n\geq 2$. By $q^{1},p_{1},\ldots,q^{n},p_{n}$ we denote the standard coordinates in $\mathbb{R}^{2n}$, and we equip $\mathbb{R}^{2n}$ with the standard symplectic form $\omega_{0}:=\sum_{i=1}^{n}dq^{i}\wedge dp_{i}$.111Following the physicists’ convention I use an upper index for the $i$-th coordinate of a point $q$ in the base manifold $\mathbb{R}^{n}$ and lower index for the $i$-th coordinate of a covector $p\in\mathbb{R}^{n}=T_{q}^{*}\mathbb{R}^{n}$. We denote by $B^{m}_{r}$ resp. ${\overline{B}}^{m}_{r}$ the open resp. closed ball in $\mathbb{R}^{m}$ of radius $r$ around 0. M. Gromov’s famous nonsqueezing theorem [Gro85, Corollary, p. 310] implies that ${\overline{B}}^{2n}_{r}$ does not symplectically embed into the closed unit symplectic cylinder ${\overline{B}}^{2}_{1}\times\mathbb{R}^{2n-2}$ if $r>1$. In [Sch03] F. Schlenk investigated how flexible symplectic embeddings are in the case $r\leq 1$. More precisely, for every $z\in\mathbb{R}^{2n-2}$ we define

[TABLE]

Answering a question by D. McDuff [McD98], in [Sch03, Theorem 1.1] Schlenk proved that for every $a>0$ there exists a symplectic embedding $\varphi$ of ${\overline{B}}^{2n}_{1}$ into ${\overline{B}}^{2}_{1}\times\mathbb{R}^{2n-2}$, such that for every $z\in\mathbb{R}^{2n-2}$ the section \iota_{z}^{-1}\big{(}\varphi\big{(}{\overline{B}}^{2n}_{1}\big{)}\big{)} has area222This means two-dimensional Lebesgue measure. at most $a$.

Schlenk’s lifting method [Sch05, Section 8.4] also shows that for every positive integer $k$ and every $a>\frac{1}{k}$ there exists a symplectic embedding of the open cube $(0,1)^{2n}$ into the open polydisc $(0,1)^{2n-1}\times(0,k)$, whose sections have area at most $a$. The main result of the present article answers the following two questions:

Question 1**.**

Is this statement true with the integer $k$ replaced by a general real number $c\geq 1$?

Question 2**.**

Can the bound $a$ on the areas of the sections be made sharp, i.e., equal to $\frac{1}{c}$?333There is always a section of area at least $\frac{1}{c}$, by Fubini’s theorem. Hence $a=\frac{1}{c}$ is the minimal possible bound.

I also answer a variant of the following question by Schlenk. For every bounded subset $S$ of $\mathbb{R}^{m}$ we define the bounded hull of $S$ to be the union of $S$ and all bounded connected components of $\mathbb{R}^{m}\setminus S$.

Question 3** (Schlenk, [Sch03], Question 2.2).**

Let $n\geq 2$, $\varphi$ be a symplectic embedding of $B^{2n}_{1}$ into $B^{2}_{1}\times\mathbb{R}^{2n-2}$, and $a<\pi$. Does there exist $z\in\mathbb{R}^{2n-2}$ such that the bounded hull of the closure of the section \iota_{z}^{-1}\big{(}\varphi\big{(}B^{2n}_{1}\big{)}\big{)} has area at least $a$?

The main result of this article is the following.

Theorem 4**.**

For every $n\geq 2$ and $c\in[1,\infty)$ there exists a symplectic embedding $\varphi:(0,1)^{2n}\to(0,1)^{2n-1}\times(0,c)$, such that for every $z\in\mathbb{R}^{2n-2}$ the following holds:

1. (i)

The section \iota_{z}^{-1}\big{(}\varphi\big{(}(0,1)^{2n}\big{)}\big{)} has area at most $\frac{1}{c}$. 2. (ii)

Its complement in $\mathbb{R}^{2}$ is path-connected.

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:

Corollary 5**.**

For every $n\geq 2$ and $a>0$ there exists a symplectic embedding $\psi:B^{2n}_{1}\to B^{2}_{1}\times\mathbb{R}^{2n-2}$, such that the bounded hull of each section of \psi\big{(}B^{2n}_{1}\big{)} has area at most $a$.

(For a proof see p. 2.) 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 ${\overline{B}}^{2n}_{1}$ and ${\overline{B}}^{2}_{1}$. Even the following is true:

Proposition 6** (F. Lalonde, D. McDuff).**

Let $n\in\mathbb{N}$ and $\varphi:{\overline{B}}^{2n}_{1}\to{\overline{B}}^{2}_{1}\times\mathbb{R}^{2n-2}$ be a symplectic embedding.444We don’t impose any restrictions on how $\varphi$ maps the boundary of the ball. Then there exists $z\in\mathbb{R}^{2n-2}$, such that the section \iota_{z}^{-1}\big{(}\varphi\big{(}{\overline{B}}^{2n}_{1}\big{)}\big{)} contains the circle of radius $1$ around 0.

In particular the bounded hull of this section equals ${\overline{B}}^{2}_{1}$, which has area $\pi$.

Proof of Proposition 6.

This follows from [LM95, Lemma 1.2]. ∎

Remark**.**

Let $\varphi$ be as in the statement of Theorem 4. Then each section of the image of $\varphi$ equals its own bounded hull. Hence $\varphi$ 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 $\Psi:(q,p)\mapsto(Q,P)$, induced by the Lagrangian shear p\mapsto P:=\big{(}p_{1},cp_{1}+p_{2}\big{)}. The $P_{2}$-sections of the image of the square $(0,1)^{2}$ under this shear have length at most $\frac{1}{c}$. Hence the area of each section of \Psi\big{(}(0,1)^{4}\big{)} is at most $\frac{1}{c}$. To make the image of $\Psi$ fit in the polydisc $(0,1)^{3}\times(0,c)$, we wrap its upper part (in $P_{2}$-direction) back to the lower part, by passing to the quotient $\mathbb{R}/c\mathbb{Z}$. We also wrap the $Q^{1}$-coordinate. See Figure 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.

Remarks** (method of proof, related result, terminology).**

• •

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 **[Tra95]** and **[Sch05, Chapter 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 **[Sch05, Section 8.3]** can be used to prove that for every $n\geq 2$, positive integer $k$, and $\ell\in(0,1)$ there exists a symplectic embedding $\varphi:(0,\ell)^{2n}\to(0,1)^{2n-1}\times(0,k)$, such that the bounded hull of every section of \varphi\big{(}(0,\ell)^{2n}\big{)} has area at most $\frac{1}{k}$. Theorem 4 improves this in the following ways:

• –

It treats the critical case $\ell=1$.

• –

It makes the area estimate sharp.

• –

It holds for any real number $c\geq 1$, 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 hull of the closure of a set its “simply connected hull”. The simply connected hull of a simply connected compact subset $S$ of $\mathbb{R}^{m}$ need not be equal to $S$. In the case $m\geq 3$ 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 $\mathbb{R}^{m}$, 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 **[Sch05]**.

2. Proofs of the main result and of Corollary 5

In the proofs of Theorem 4 and Corollary 5 we will use the following lemma.

Lemma 7** (squaring the disc and the cylinder).**

We denote $r:=\pi^{-\frac{1}{2}}$.

1. (i)

There exists a homeomorphism

[TABLE]

that restricts to a (smooth) symplectomorphism between the interiors. 2. (ii)

For every $y_{0}\in(0,1)^{2}$ there exists continuous map

[TABLE]

that maps $(\mathbb{R}/\mathbb{Z})\times\{1\}$ to $y_{0}$, and restricts to a homeomorphism from $(\mathbb{R}/\mathbb{Z})\times[0,1)$ to $[0,1]^{2}\setminus\{y_{0}\}$ and to a symplectomorphism from $(\mathbb{R}/\mathbb{Z})\times(0,1)$ to $(0,1)^{2}\setminus\{y_{0}\}$.

The idea of proof of this lemma is explained by Figure 2.

In the proof of Lemma 7 we will use the following.

Remark 8** (straightening corners).**

We denote by $\Sigma$ the square $[0,1]^{2}$ without the corners. Let $r>0$ and $S$ be a subset of the circle of radius $r$ consisting of four points. There exists homeomorphism $\theta:[0,1]^{2}\to{\overline{B}}^{2}_{r}$ that restricts to a diffeomorphism from $\Sigma$ onto ${\overline{B}}^{2}_{r}\setminus S$, such that $(\theta|\Sigma)_{*}\omega_{0}$ extends to a nonvanishing smooth 2-form on ${\overline{B}}^{2}_{r}$.

To see this, observe that the map

[TABLE]

is a homeomorphism that restricts to a diffeomorphism from $[0,\infty)^{2}\setminus\{0\}$ onto \big{(}\mathbb{R}\times[0,\infty)\big{)}\setminus\{0\}, that satisfies

[TABLE]

The desired map $\theta$ can be constructed from four copies of ${\widetilde{\theta}}$ (one for each corner), using charts for ${\overline{B}}^{2}_{r}$ and a cut off argument.

Proposition 9** (Banyaga’s Moser stability with boundary).**

Let $M$ be a compact connected oriented smooth manifold, and $\Omega_{0},\Omega_{1}$ volume forms on $M$ satisfying

[TABLE]

Then there exists diffeomorphism $\varphi$ of $M$ satisfying

[TABLE]

Proof.

See [Ban74, Théorème, p. 127]. ∎

Proof of Lemma 7.

To prove (i), we define $r:=\pi^{-\frac{1}{2}}$ and choose a map $\theta$ as in Remark 8. We define

[TABLE]

We have

[TABLE]

Hence the hypotheses of Proposition 9 are satisfied. We choose a diffeomorphism $\varphi$ as in the statement of this proposition. The map

[TABLE]

has the required properties.

We prove (ii). There exists a symplectomorphism

[TABLE]

For example, consider $y:\mathbb{R}/\mathbb{Z}\to\mathbb{C}=\mathbb{R}^{2}$, $y(\bar{q}):=e^{2\pi iq}$, where $q\in\bar{q}$ is an arbitrary representative, and define

[TABLE]

We choose a symplectomorphism $\xi$ of $[0,1]^{2}$ that equals the identity in a neighbourhood of the boundary and maps $\kappa(0)$ to $y_{0}$.555$\xi$ is a smooth map in the sense of manifolds with boundary and corners. We obtain such a map as the Hamiltonian flow of a suitable function on $(0,1)^{2}$ with compact support. The map

[TABLE]

has the required properties. This proves (ii) and completes the proof of Lemma 7. ∎

Proof of Theorem 4.

Consider the case $n=2$. We denote by

[TABLE]

the canonical projection, and equip $(\mathbb{R}/\mathbb{Z})\times\mathbb{R}\times\mathbb{R}\times(\mathbb{R}/c\mathbb{Z})$ with the symplectic form induced by $\omega_{0}$ and $\pi$. We denote $y_{0}:=z_{0}:=\left(\frac{1}{2},\frac{1}{2}\right)$. We choose a map $\lambda$ as in Lemma 7(ii). It follows from the same lemma that there exists a symplectomorphism

[TABLE]

We define

[TABLE]

The map $\varphi$ is well-defined, since $\pi\circ\Psi$ maps $(0,1)^{4}$ to the product of the domains of $\lambda$ and $\lambda^{\prime}$. The map $\varphi$ is a symplectic immersion, as it is the composition of three symplectic immersions. A straight-forward argument shows that \pi\circ\Psi\big{|}(0,1)^{4} is injective. Since $\lambda|(\mathbb{R}/\mathbb{Z})\times(0,1)$ and $\lambda^{\prime}$ are injective, it follows that the same holds for $\varphi$. Hence $\varphi$ is a symplectic embedding of $(0,1)^{4}$ into $(0,1)^{3}\times(0,c)$.

Let $(Q^{2},{\overline{P}}_{2})\in(0,1)\times(\mathbb{R}/c\mathbb{Z})$. We have

[TABLE]

[TABLE]

[TABLE]

where $\frac{{\overline{P}}_{2}-p_{2}}{c}\in\mathbb{R}/\mathbb{Z}$. The set $W_{{\overline{P}}_{2}}$ is an open subinterval of $(0,1)$ or the union of two such subintervals. It has length $\frac{1}{c}$. Using (1) and (2), it follows that $U_{Q^{2},{\overline{P}}_{2}}$ has area equal to $\frac{1}{c}$. Figure 3 depicts the set $U_{Q^{2},{\overline{P}}_{2}}$.

Let now z\in\big{(}(0,1)\times(0,c)\big{)}\setminus\{z_{0}\}. We denote $(Q^{2},{\overline{P}}_{2}):={\lambda^{\prime}}^{-1}(z)$. We have

[TABLE]

Since $\lambda$ is area-preserving, it follows that the section \iota_{z}^{-1}\big{(}\varphi\big{(}(0,1)^{4}\big{)}\big{)} has area equal to $\frac{1}{c}$. For $z=z_{0}$ or $z$ outside of $(0,1)\times(0,c)$, the section is empty. This proves (i).

To prove property (ii), consider the continuous path

[TABLE]

The point $y(0)$ lies on the boundary of the square $[0,1]^{2}$. It follows from (2) that the path $y$ lies inside the complement of \iota_{z}^{-1}\big{(}\varphi\big{(}(0,1)^{4}\big{)}\big{)} in $\mathbb{R}^{2}$. Every point outside $(0,1)^{2}$ can be connected to $y(0)$ through a continuous path outside of $(0,1)^{2}$. Every point in the complement of \iota_{z}^{-1}\big{(}\varphi\big{(}(0,1)^{4}\big{)}\big{)} in $(0,1)^{2}$ can be connected to a point on the path $y$ through a path in this complement. This follows from (4) and the facts $U_{Q^{2},{\overline{P}}_{2}}=V_{Q^{2}}\times W_{{\overline{P}}_{2}}$, $V_{Q^{2}}=(\mathbb{R}/\mathbb{Z})\setminus\{-cQ^{2}+\mathbb{Z}\}$. See again Figure 3. This proves (ii).

Hence $\varphi$ has the desired properties. This proves Theorem 4 in the case $n=2$. For $n\geq 3$ we take the product of $\varphi$ with the identity map. ∎

In the proof of Corollary 5 we will use the following.

Remark 10** (monotonicity).**

The bounded hull is monotone in the sense that if $A\subseteq B\subseteq\mathbb{R}^{m}$ are bounded sets then the bounded hull of $A$ is contained in the bounded hull of $B$.

Proof of Corollary 5.

We define $r:=\pi^{-\frac{1}{2}}$. By a rescaling argument it suffices to show that for every $a\in(0,1]$ there exists a symplectic embedding $\psi:B^{2n}_{r}\to B^{2}_{r}\times\mathbb{R}^{2n-2}$, such that the bounded hull of each section of $\psi(B^{2n}_{r})$ has area at most $a$. To prove this statement, we choose $\varphi$ is as in the conclusion of Theorem 4 with $c:=\frac{1}{a}$. We choose a map $\kappa$ as in Lemma 7(i). The map

[TABLE]

is a symplectic embedding. Let $z\in\mathbb{R}^{2n-2}$. Property (ii) in Theorem 4 implies that the complement of V:=\kappa^{-1}\Big{(}\iota_{z}^{-1}\big{(}\varphi\big{(}(0,1)^{2n}\big{)}\big{)}\Big{)} in $\mathbb{R}^{2}$ is path-connected. Hence $V$ equals its bounded hull. The section \iota_{z}^{-1}\big{(}\psi(B^{2n}_{r})\big{)} is contained in $V$. Using Remark 10, it follows that the bounded hull of this section is also contained in $V$. Using Theorem 4(i) and that $\kappa$ is area-preserving, it follows that this bounded hull has area at most $\frac{1}{c}=a$. Hence $\psi$ has the desired properties. This proves Corollary 5. ∎

3. Acknowledgments

I would like to thank Felix Schlenk for an interesting discussion and for proof-reading the first version of this article.