LCS Non-Squeezing Theorem

• Lcs non-squeezing theorem is a rigidity result in locally conformally symplectic geometry that generalizes Gromov's theorem by using a non-degenerate 2-form with a non-exact Lee form.
• It employs deformation theory, Gromov–Witten invariants, and generating function techniques to derive integer-valued capacities and characterize embedding obstructions.
• The theorem bridges symplectic and contact geometries, highlighting practical embedding constraints and open problems in non-exact lcs deformations.

A locally conformally symplectic (lcs) non-squeezing theorem is a rigidity phenomenon generalizing the famous Gromov non-squeezing theorem from symplectic to locally conformally symplectic geometry. In contrast to symplectic manifolds, which admit a global closed non-degenerate 2-form, an lcs manifold admits a non-degenerate 2-form $\omega$ satisfying $d\omega = \eta\wedge\omega$ for some closed 1-form $\eta$ (the Lee form). This subtle weakening leads to new obstructions and technical differences both in the definition of capacities and in the structure of possible embeddings and isotopies. The lcs non-squeezing problem seeks to characterize which lcs (or symplectic) embeddings are possible between standard “balls” and “cylinders” in lcs manifolds, and how such phenomena interpolate between symplectic and contact geometries through the presence of the Lee form.

1. Locally Conformally Symplectic Structures and Embeddings

A lcs manifold $M^{2n}$ is a smooth manifold equipped with a nondegenerate 2-form $\omega$ and a closed 1-form $\eta$ with $d\omega = \eta\wedge\omega$. Locally, $\omega$ can be written as $e^{f_U}\omega_U$ where $\omega_U$ is symplectic and $f_U\in C^\infty(U)$. The pair $(\eta, \omega)$ is defined up to the equivalence $(\eta',\omega') = (\eta + df, e^{f}\omega)$.

A smooth embedding $\phi: M \to M'$ between lcs manifolds $(M, \omega, \eta)$ and $(M', \omega', \eta')$ is an lcs embedding if there exists a positive function $f$ such that $\phi^*\omega' = f\omega$ and $\phi^*\eta' = \eta + d(\ln f)$ (Savelyev, 2022, Bertelson et al., 20 Nov 2025). When the Lee form is exact, lcs structures reduce globally to conformally symplectic, recovering the symplectic case.

2. Statement of the lcs Non-Squeezing Theorems

2.1. Deformation-Theoretic Non-Squeezing in lcs Manifolds

For the model $M = S^2 \times T^{2n-2}$ with the product symplectic form $\omega_0$, the standard ball $B_R \subset \mathbb{R}^{2n}$ is considered. The lcs non-squeezing theorem (Savelyev, 2022) states:

Let $R>r>0$. There exists $\epsilon>0$ such that for any continuous path $\{\omega_t\}_{t\in[0,1]}$ of lcs forms on $M$ with $\omega_0 = \omega_0$ and

$\max_t d_0(\omega_t,\omega_0)<\epsilon$

(in a suitable $C^0$-topology), there does not exist a smooth embedding $\phi: B_R \to M$ such that $\phi^*\omega_1 = \omega_{\mathrm{st}}$ and, where relevant, $\phi_*$ is leaf-preserving on certain hypersurfaces. This prohibits symplectic embedding of standard balls of radius $R$ into $(U, \omega_1)$, where $$U = M \setminus \bigcup_i \Sigma_i \cong S^2\times\mathbb{R}^{2n-2}$$, for any small lcs deformation. In the limit $\omega_t\equiv\omega_0$, this recovers the classical Gromov non-squeezing.

2.2. Integer-Valued Non-Squeezing and lcs Capacity

For certain lcs spaces such as $M = S^1 \times \mathbb{R}^{2n} \times S^1$ with $(\eta,\omega)=(-d\theta, d_{-d\theta}\alpha_0)$, the lcs non-squeezing theorem manifests via an integer-valued capacity (Bertelson et al., 20 Nov 2025):

For any $k\in\mathbb{N}_0$ and radii $R_1, R_2 >0$ with

$\pi R_2^2 \leq k \leq \pi R_1^2,$

there does not exist a compactly supported lcs Hamiltonian diffeomorphism $\varphi$ such that

$$\varphi(\overline{S^1 \times B^{2n}(R_1) \times S^1}) \subset S^1 \times B^{2n}(R_2) \times S^1.$$

Equivalently, in terms of the lcs capacity $c(\cdot)$,

$$c(S^1\times B^{2n}(R_1)\times S^1) = \lceil \pi R_1^2\rceil > k \geq \lceil \pi R_2^2\rceil = c(S^1\times B^{2n}(R_2)\times S^1).$$

3. Proof Strategies and Technical Framework

3.1. Deformation and Gromov–Witten Methods

The deformation-theoretic approach exploits $C^0$-smallness of the lcs perturbation, the construction of almost complex structures preserving relevant foliations, positivity of intersections, and Gromov–Witten invariants. On the universal cover, each lcs form is written as $f_t\omega_t^{\mathrm{symp}}$ with $\omega_t^{\mathrm{symp}}$ closed. Mutual closeness yields uniform bounds, which facilitate compactness via Gromov–Witten theory and prevent holomorphic curves from “escaping to infinity” (Savelyev, 2022).

3.2. Spectral Selector and Generating Function Approach

For domains modeled on $S^1\times\mathbb{R}^{2n}\times S^1$, lcs Hamiltonian dynamics are encoded by spectral selectors associated to generating functions quadratic at infinity for Lagrangian submanifolds in twisted cotangent bundles. For $\varphi\in\mathrm{Ham}^c(M)$, the selectors

$\ell_+(\varphi),\ \ell_-(\varphi)$

arise as action selectors via the generating function, supporting an integer-valued lcs capacity and bi-invariant metric. The proof uses monotonicity, triangle inequalities, displacement estimates, and careful calculation of the integer capacity for standard balls (Bertelson et al., 20 Nov 2025).

4. Relationship with Symplectic and Contact Non-Squeezing

The classical Gromov non-squeezing theorem asserts that no Hamiltonian symplectic embedding can squeeze $B^{2n}(R_1)$ into the cylinder $Z^{2n}(R_2) = B^2(R_2) \times \mathbb{R}^{2n-2}$ unless $\pi R_1^2 \leq \pi R_2^2$. In the contact setting, Eliashberg–Kim–Polterovich (EKP) non-squeezing for $\mathbb{R}^{2n}\times S^1$ implies similar constraints, but with integer-valued capacities due to periodicity of the Reeb flow: a "quantum gap" exists at integer thresholds.

The lcs non-squeezing theorems interpolate these properties. Like in the contact case, the Lee flow gives rise to periodicity and integer capacities, and like in the symplectic case, rigid partial orders and generating function invariants are central. The forbidden region for ball embeddings occurs at integer values of $\pi R^2$ (Bertelson et al., 20 Nov 2025).

5. Corollaries, Special Cases, and Examples

For $n=1$, $M=S^1\times\mathbb{R}^2\times S^1$, the standard lcs ball is $S^1\times B^2(R)\times S^1$. When $\pi R^2<1$, embedding into arbitrarily small lcs cylinders is permitted; for $1\leq \pi R_2^2\leq \pi R_1^2$, non-squeezing rigidity holds as in the EKP case (Bertelson et al., 20 Nov 2025). In the globally conformal symplectic case (exact Lee form), the theorems reduce to conformally invariant versions of Gromov’s result.

A conjectural lcs analogue of contact non-squeezing proposes that for $R \geq 1$ there is no compactly supported Hamiltonian lcs map that squeezes $\overline{B_R} \times S^1 \times S^1$ into $B_R \times S^1 \times S^1$, generalizing EKP non-squeezing to the lcs context (Savelyev, 2022).

6. Open Problems and Future Directions

Key open questions include: the classification of non-exact lcs deformations (beyond exact or conformal cases), the extension of deformation non-squeezing to broader classes of symplectic manifolds with nontrivial $H^1$, the removal of technical restrictions like leaf-preservation, and the formulation of a full lcs Gromov–Witten theory supporting virtual cycles and non-squeezing without intersection-theoretic assumptions (Savelyev, 2022). The existence of essential Lee chords and translated points in Liouville lcs manifolds, and precise analogues of the Lagrangian and Legendrian Arnold conjectures, remain active research areas (Bertelson et al., 20 Nov 2025).

7. Comparisons and Summary Table

A summary of the non-squeezing phenomena across the symplectic, contact, and lcs settings:

Setting	Capacity	Thresholds	Embedding Obstruction
Symplectic ($\mathbb{R}^{2n}$)	$c_{\rm Viterbo}(B(R)) = \pi R^2$	$\pi R_1^2 > \pi R_2^2$	No symplectic embedding of $B^{2n}(R_1)$ into $Z^{2n}(R_2)$
Contact ($\mathbb{R}^{2n}\times S^1$)	$c_{\rm EKP}(B(R)\times S^1)=\lceil\pi R^2\rceil$	$\pi R_1^2 \ge 1 \ge \pi R_2^2$	No contact isotopy squeezing for integer threshold
lcs ($S^1 \times \mathbb{R}^{2n} \times S^1$)	$c(S^1\times B(R)\times S^1)=\lceil\pi R^2\rceil$	Integer $k$, $\pi R_2^2 \le k \le \pi R_1^2$	No compactly supported lcs Hamiltonian isotopy squeezing at integer gap

The lcs non-squeezing theorem thus represents a natural and rigorous generalization, interpolating geometric rigidity between the strictly symplectic and contact regimes, and is underpinned by new invariants, capacities, and deformation principles (Savelyev, 2022, Bertelson et al., 20 Nov 2025).