A remark on deformation of Gromov non-squeezing

TL;DR

This paper explores the persistence of Gromov non-squeezing under small deformations of the symplectic form, providing partial proofs in specific cases and introducing a novel trap method for holomorphic curves.

Contribution

It conjectures the stability of Gromov non-squeezing under $C^0$-small symplectic form deformations and proves this in certain low-dimensional cases.

Findings

Gromov non-squeezing persists under specific deformations

Introduces a trap method for holomorphic curves

Proves conjecture in dimension four when R < √2 r

Abstract

Let $R,r$ be as in the classical Gromov non-squeezing theorem, and let $\epsilon = (\pi R ^{2} - \pi r ^{2})/ \pi r ^{2} $. We first conjecture that the Gromov non-squeezing phenomenon persists for deformations of the symplectic form on the range $C ^{0}$ (w.r.t. the standard metric) $\epsilon $-nearby to the standard symplectic form. We prove this in some special cases, in particular when the dimension is four and when $R < \sqrt 2 r$. Given such a perturbation, we can no longer compactify the range and hence the classical Gromov argument breaks down. Our main method consists of a certain trap idea for holomorphic curves, analogous to traps in dynamical systems.

Equations

$$(B_R,{\omega }_{st})\hookrightarrow (S^2\times {\mathbb{R}}^{2n-2},{\omega }_{\pi r^2}\oplus {\omega }_{st}),$$

$$\mathrm{image}u\cap K\ne \varnothing ⟹\mathrm{image}u\subset K.$$

$$d_{C^0}({\omega }_0,{\omega }_1)=\underset{|v\wedge w{|}_g=1}{\sup }|{\omega }_0(v,w)-{\omega }_1(v,w)|,$$

$$\varphi :B_R\hookrightarrow (M,{\omega }_1).$$

$$|⟨{\omega }_1,A⟩-\pi \cdot r^2|=|⟨{\omega }_1,A⟩-⟨\omega ,A⟩|\le ϵ\pi \cdot r^2,$$

$$|{\int }_{{\mathbb{CP}}^1}{u}_0^{\ast }{\omega }_1-\pi r^2|<\pi R^2-\pi r^2,$$

$${\int }_{{\mathbb{CP}}^1}{u}_0^{\ast }{\omega }_1<\pi R^2.$$

Full text

1. Introduction

One of the most important to this day results in symplectic geometry is the so called Gromov non-squeezing theorem, appearing in the seminal paper of Gromov [3]. Let $\omega_{st}=\sum_{i=1}^{n}dp_{i}\wedge dq_{i}$ denote the standard symplectic form on $\mathbb{R}^{2n}$. Gromov’s theorem then says that there does not exist a symplectic embedding

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for $R>r$, with $B_{R}$ the standard closed radius $R$ ball in $\mathbb{R}^{2n}$ centered at [math], and $\omega_{\pi r^{2}}$ a symplectic form on $S^{2}$ with area $\pi r^{2}$.

We show that in dimension 4 Gromov’s non-squeezing is $C^{0}$ persistent in the following sense.

Theorem 1.1**.**

Let $R>r>0$ be given, and let $\omega=\omega_{\pi r^{2}}\oplus\omega_{st}$ be the symplectic form on $M=S^{2}\times\mathbb{R}^{2}$ as above. Then there is an $\epsilon>0$ s.t. for any symplectic form $\omega^{\prime}$ on $M$, $C^{0}$ $\epsilon$-close to $\omega$, there is no symplectic embedding $\phi:(B_{R},\omega_{st})\hookrightarrow(M,\omega^{\prime})$, meaning that $\phi^{*}\omega^{\prime}=\omega_{st}$.

It is not clear if the dimension 4 restriction is essential. A suitable holomorphic trap (Definition 2.1) is certainly much more difficult to construct in higher dimensions.

It is natural to ask if the above theorem continues to hold for general nearby forms. Or formally this translates to:

Question 1*.*

Let $R>r>0$ be given, and let $\omega=\omega_{\pi r^{2}}\oplus\omega_{st}$ be the symplectic form on $M=S^{2}\times\mathbb{R}^{2n-2}$, as above. For every $\epsilon>0$ is there a (necessarily globally non-closed, for very small $\epsilon$) 2-form $\omega^{\prime}$ on $M$, $C^{0}$ $\epsilon$-close to $\omega$, such that there is a symplectic embedding $\phi:B_{R}\hookrightarrow M$, i.e. s.t. $\phi^{*}\omega^{\prime}=\omega_{st}$?

We cannot readily reduce this question to just applying Theorem 1.1 (in dimension 4). This is because, while a symplectic form on a subdomain of the form $\phi(B_{R})\subset M$ extends to a symplectic form, by a classical theorem of Gromov  [2], the extension may not be $C^{0}$ close. Indeed, this appears to be rather unlikely to happen, unless there exists a stronger type of h-principle than Gromov’s which allows us to control the uniform norm.

The above question seems to be difficult. My opinion is that the answer is ‘yes’, in part because it is difficult to imagine any obstruction, for example we no longer have Gromov-Witten theory for general $\omega^{\prime}$. On the other, my attempts to construct an example failed, so that ‘no’ is certainly very possible.

A work of Müller [6] explores a different kind of question, by instead relaxing the condition of the map being symplectic. This is a very different idea, and there is no direct connection to our problem, as pull-backs by diffeomorphisms of nearby forms may not be nearby. Hence, there is no way to go from nearby embeddings that we work with to $\epsilon$-symplectic embeddings of Müller.

2. A trap for holomorphic curves

For basic notions of $J$-holomorphic curves we refer the reader to [5].

Definition 2.1**.**

Let $(M,J)$ be an almost complex manifold, and $A\in H_{2}(M)$ fixed. Let $K\subset M$ be a closed subset. Suppose that for every $x\in\partial K$ (the topological boundary) there is a ${J}$-holomorphic (real codimension $2$) compact hyperplane $H_{x}$ through $x$ satisfying:

• •

$H_{x}\subset K$.

• •

$A\cdot H_{x}\leq 0$, where the left-hand side is the homological intersection number.

We call such a $K$ a $J$-holomorphic trap (for class $A$ curves).

Lemma 2.2**.**

Let $M,J$ and $A$ be as above, and $K$ be a $J$-holomorphic trap for class $A$ curves. Let $u:\Sigma\to M$ be a $J$-holomorphic class $A$ curve $u$ with $\Sigma$ a connected closed Riemann surface. Then

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Proof.

Suppose that ${u}$ intersects $\partial K$, otherwise we already have $\operatorname{\mathrm{image}}{u}\subset interior(K)$, since $\operatorname{\mathrm{image}}{u}$ is connected (and by elementary topology). Then ${u}$ intersects $H_{x}$ as in the definition of a holomorphic trap, for some $x$. Consequently, as $A\cdot H_{x}\leq 0$, by positivity of intersections [5, Section 2.6], $\operatorname{\mathrm{image}}u\subset H_{x}\subset K$. ∎

3. Proof of Theorem 1.1

Definition 3.1**.**

We say that a pair $(\omega,J)$ of a 2-form $\omega$ on $M$ and an almost complex structure $J$ on $M$ are compatible if $\omega(\cdot,J\cdot)$ defines a $J$-invariant inner product $g_{\omega,J}$ on $M$.

Let $\omega$ be the symplectic form on $M=S^{2}\times\mathbb{R}^{2}$ as in the statement. Let us quickly recall the definition of the $C^{0}$ distance $d_{C^{0}}$, on the set of 2-forms $\Omega^{2}(M)$ for a fixed metric $g$ on $M$.

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where more specifically, the supremum is over all $g$-norm 1 bivectors $v\wedge w$ in $\Lambda^{2}(TM)$. In our case $d_{C^{0}}$ will be defined with respect to the metric $g_{\omega,J}$ as in Definition 3.1 for $J$ the standard product complex structure.

Suppose by contradiction that for every $\epsilon>0$ there is an $\omega_{1}$ s.t. $d_{C^{0}}(\omega,\omega_{1})<\epsilon$ and such that there exists a symplectic embedding

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Let $\epsilon^{\prime}>0$ be s.t. any symplectic form $\omega_{1}$ on $M$, $C^{0}$ $\epsilon^{\prime}$-close to $\omega$ satisfies:

• •

$\omega_{t}=(1-t)\omega+t\omega_{1}$ is non-degenerate, for each $t\in[0,1]$.

• •

For each $t\in[0,1]$, $\omega_{t}$ is non-degenerate on all the fibers of the natural projection $p:(M=S^{2}\times\mathbb{R}^{2})\to\mathbb{R}^{2}$. In what follows we just call them fibers.

For $\epsilon<\epsilon^{\prime}$ as above, let $\omega_{1}$ and $\phi:B_{R}\to(M,\omega_{1})$ be as in our hypothesis. Set $B:=\phi(B_{R})$ and let $D\supset B$ be an open domain, with compact closure $K$, s.t. $K$ is the product $S^{2}\times D^{2}$ for $D^{2}\subset\mathbb{R}^{2}$ a closed disk. In particular, $\partial K$ is smoothly folliated by the fibers. We denote by $T^{vert}\partial K\subset TM$, the sub-bundle of vectors tangent to the leaves of the above-mentioned foliation.

We may extend $\phi_{*}j$ to an $\omega_{1}$-compatible almost complex structure $J_{1}$ on $M$, preserving $T^{vert}\partial K$ using:

• •

$\operatorname{\mathrm{image}}\phi$ does not intersect $\partial K$.

• •

The non-degeneracy of $\omega_{1}$ on the fibers, which follows by the defining condition of $\epsilon$.

• •

The well known existence/flexibility results for compatible almost complex structures on symplectic vector bundles.

We may then extend $J_{1}$ to a family $\{J_{t}\}$, $t\in[0,1]$, of almost complex structures on $M$, s.t. $J_{t}$ is $\omega_{t}$-compatible for each $t$, with $J_{0}=J$ as above, and such that $J_{t}$ preserves $T^{vert}\partial K$ for each $t$. The latter condition can be satisfied by similar reasoning as above, using that $\omega_{t}$ is non-degenerate on the fibers for each $t$.

So the fibers above are $J_{t}$-holomorphic hyperplanes for each $t$, and smoothly foliate $\partial K$. Moreover, if $A=[S^{2}]\otimes[pt]$ is as in the statement, then the intersection number of $A$ with a fiber is 0. That is $A\cdot p^{-1}(z)=0$, for $\forall z\in\mathbb{R}^{2}$. And so $K$ is a compact $J_{t}$-holomorphic trap for class $A$ curves, for each $t$.

Set $x_{0}:=\phi(0)$. Denote by $\mathcal{M}_{t}$ the space of equivalence classes of maps $u:\mathbb{CP}^{1}\to M$, where $u$ is a $J_{t}$-holomorphic, class $A$ curve passing through $x_{0}$. The equivalence relation is by the usual biholomorphism reparametrization group action, so that $u\sim u^{\prime}$ if there exists a biholomorphism $\phi:\mathbb{CP}^{1}\to\mathbb{CP}^{1}$ s.t. $u^{\prime}=u\circ\phi$. Then $\mathcal{M}=\cup_{t}\mathcal{M}_{t}$ is compact by energy minimality of $A$ (which rules out bubbling), by Lemma 2.2, and by compactness of $K$.

As explained in  [1, Section 3.5], in a essentially identical situation, we may embed $\mathcal{M}$ into a natural polyfold setup of Hofer-Wysocki-Zehnder  [4]. That is we express $\mathcal{M}$ as the zero set of an sc-Fredholm section of a suitable (tame, strong) $M$-polyfold bundle. The only difference with their setup is that they compactify $M$ to $S^{2}\times T^{2}$. We of course cannot compactify, and so we have to use the holomorphic trap idea, to force compactness of $\mathcal{M}$. Again as in  [1], we take the $M$-polyfold regularization of $\mathcal{M}$. This gives a one dimensional compact cobordism $\mathcal{M}^{reg}$ between $\mathcal{M}^{reg}_{0}$ and $\mathcal{M}^{reg}_{1}$.

Now $\mathcal{M}^{reg}_{0}$ is a point: corresponding to the unique $(J=J_{0})$-holomorphic class $A$, curve $u:\mathbb{CP}^{1}\to M$ passing through $x_{0}$. Consequently, $\mathcal{M}^{reg}_{1}$ is non-empty, that is there is a $J_{1}$-holomorphic class $A$ curve $u_{0}:\mathbb{CP}^{1}\to M$ passing through $x_{0}$.

Remark 3.2*.*

It is certainly possible that more classical, geometric perturbation style arguments may be adopted to the present problem. There are however difficulties: it is important for us to work with curves constrained to pass through a specific point, instead of doing homological intersection of an unconstrained evaluation cycle, with a point (as in the classical proof of Gromov non-squeezing). For without the specific constraint our moduli space is not even compact, and hence the homological intersection theory makes no sense. Such a constraint may not neatly fit into classical analytical framework of McDuff-Salamon  [5].

Now we have:

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as $\langle{\omega},A\rangle=\pi r^{2}$, and as $d_{C^{0}}({\omega},{\omega}_{1})<\epsilon$, (also using that we can find a representative for $A$ whose $g$-area is $\pi r^{2}$). So choosing $\epsilon$ appropriately we get

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And consequently,

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We may then proceed exactly as in the now classical proof of Gromov [3] of the non-squeezing theorem to get a contradiction and finish the proof. A bit more specifically, $\phi^{-1}({\operatorname{\mathrm{image}}\phi\cap\operatorname{\mathrm{image}}u_{0}})$ is a minimal surface in $B_{R}$, with boundary on the boundary of $B_{R}$, and passing through $0\in B_{R}$. By construction it has area strictly less than $\pi R^{2}$, which is impossible by the classical monotonicity theorem of differential geometry. See also [1] where the monotonicity theorem is suitably generalized, to better fit the present context. ∎

4. Acknowledgements

I am grateful to Felix Schlenk, Misha Gromov and Dusa McDuff for some feedback.