Long Cylinders in Legendrian Cobordisms

• Long cylinders over Legendrian submanifolds are exact Lagrangian models forming the symplectization of Legendrians in contact manifolds.
• They serve as fundamental building blocks for constructing Lagrangian cobordisms and analyzing Legendrian isotopies in symplectic field theory.
• Their study combines flexible Hamiltonian isotopies with rigid analytical, topological, and sheaf-theoretic obstructions to embedding and non-squeezing phenomena.

A long cylinder over a Legendrian submanifold is the standard model for an exact, embedded Lagrangian cylinder or "symplectization" of a Legendrian in a contact manifold. This construction plays a foundational role in the topology of Lagrangian cobordisms, the structure of Symplectic Field Theory (SFT), and the classification of Legendrian submanifolds via their symplectizations. The theory encompasses both the flexible aspects—such as the existence of Hamiltonian isotopies between inequivalent Legendrians via long cylinders—and the rigid phenomena arising from analytic, topological, and sheaf-theoretic obstructions to the embeddability and length of such cylinders.

1. Construction and Models of Long Cylinders

Let $(Y,\alpha)$ be a $(2n-1)$-dimensional contact manifold with contact form $\alpha$, and $(\mathbb{R}_t \times Y, d(e^t\alpha))$ its symplectization. A Legendrian submanifold $\Lambda \subset Y$ admits an associated exact Lagrangian cylinder

$$L_\Lambda = \mathbb{R}_t \times \Lambda \subset \mathbb{R} \times Y,$$

with the Liouville form $e^t \alpha$ restricting to an exact form (trivially, since $\alpha|_\Lambda = 0$ by the Legendrian condition). More generally, an exact Lagrangian cobordism from a Legendrian $\Lambda_-$ to another Legendrian $\Lambda_+$ is an embedded, orientable $L \subset \mathbb{R} \times Y$ agreeing with cylinders over $\Lambda_-$ for $t \ll 0$ and over $\Lambda_+$ for $t \gg 0$, that is,

$$L \cap ((-\infty, t_-] \times Y) = (-\infty, t_-] \times \Lambda_-, \quad L \cap ([t_+, \infty) \times Y) = [t_+, \infty) \times \Lambda_+,$$

for some $t_- < t_+$, with $L$ cylindrical outside $[t_-, t_+]$ (Sabloff et al., 2015, Ekholm et al., 2012).

Long cylinders arise naturally both as standalone objects (modeling the symplectization) and as the "neck" regions in more general cobordisms, where stretching the neck isolates holomorphic and Floer-theoretic contributions.

2. Flexibility and Rigidity: Length, Capacities, and Cobordism

The notion of the "length" $\ell(L)$ of a Lagrangian cobordism $L$ is the infimum of $t_+ - t_-$ such that $L$ is cylindrical outside $[t_-, t_+]$. A central result is that while expansions ("vertical dilations" in the Reeb direction) can be realized by arbitrarily short cobordisms, contractions are rigid and yield a sharp lower bound on $\ell(L)$, measurable through filtered Legendrian Contact Homology capacities. Explicitly, for classes $\theta \in LCH^*(\Lambda_-, \epsilon_-)$ and their push-forward under the $A_\infty$ cobordism maps, the monotonicity inequality holds:

$$\ell(L) \geq \ln (c_k^-(\Lambda_-,\epsilon_-;\theta)) - \ln (c_k^+(\Lambda_+,\epsilon_+;\Psi_k^{L,\epsilon_-}(\theta))),$$

with $c_k^\pm$ denoting capacities derived from the action filtration on Reeb chords. This captures the symplectic "cost" of moving between ends, with particular rigidity in contractions and lower bounds arising from linking phenomena or non-trivial Legendrian loops (Sabloff et al., 2015).

For vertically scaled Legendrians $\Lambda_T := \phi_T(\Lambda)$ under the Reeb-$\mathbb{R}$ action, a cobordism exists with

$\ell(L_T) \geq |T|,$

while for $T>0$ (expansion), no positive lower bound is enforced. Linking two Legendrians with different shifts imposes logarithmic lower bounds on the joint cobordism length.

3. Neck-Stretching, Functoriality, and Holomorphic Curve Compactness

The insertion of arbitrarily long cylinders—"neck-stretching"—between cylindrical ends in cobordisms is essential for decoupling holomorphic curve contributions in SFT. As the neck length $T\to\infty$, moduli spaces of holomorphic disks "break," achieving compactified limits essential for gluing techniques. This mechanism also ensures that the induced dg-algebra maps on Legendrian contact homology become functorial with respect to composition of cobordisms:

$\Phi_{L_{02}} = \Phi_{L_{12}} \circ \Phi_{L_{01}},$

with the long cylinder interpolating between cobordisms. These structures underlie the TQFT-type invariants built from Legendrian contact homology and are critical for the construction of augmentation push-forwards and category-theoretic operations in SFT (Ekholm et al., 2012).

4. Isotopy, Equivalence, and Loss of Invariants under Long Cylinders

In high-dimensional settings ($2n-1 \geq 11$), pairs of Legendrian submanifolds which are not diffeomorphic, and may not even be smoothly equivalent, can possess Hamiltonian-isotopic Lagrangian cylinders in the symplectization. The key mechanism is the construction of invertible exact Lagrangian cobordisms (via flexible h-cobordisms) and the application of the Mazur trick: two Legendrians $\Lambda_0$, $\Lambda_1$ are related by invertible cobordisms if and only if their cylinders $L_{\Lambda_0}$, $L_{\Lambda_1}$ are Hamiltonian isotopic through a compactly supported isotopy (Courte, 2015).

This implies that the symplectization loses fine invariants distinguishing Legendrians when $\mathbb{R}$-equivariance is forgotten. Explicit handleslide and flexibility arguments ensure invertibility—so that even non-diffeomorphic Legendrians can have indistinguishable symplectized images up to Hamiltonian isotopy. The phenomenon is robust above the middle dimension, and examples include instances involving lens spaces and their flexible h-cobordisms.

5. Symplectic and Contact Barriers: Non-squeezing and Categorical Obstructions

Quantitative rigidity for long cylinders over Legendrians is enforced by global symplectic and categorical barriers. In prequantization bundles over integral symplectic manifolds, the existence of explicit Legendrian barriers $\Lambda_k$—constructed as lifts of Weinstein skeleta—obstructs the embedding of long Reeb or contact cylinders over arbitrary Legendrians. Once such a barrier is present, no cylinder of height $>1/k$ (measured by the Reeb flow) can embed without either intersecting $\Lambda_k$ or developing chords of length $\leq 1/k$ (Opshtein, 21 Dec 2025).

Symplectic topology and sheaf-theoretic methods extend this rigidity. For Legendrian lifts of Lagrangians in prequantized cylinders, Legendrian non-squeezing results state that the radius $R$ of a prequantized cylinder containing $\Lambda$ must satisfy $\pi R^2 \geq A_{\mathrm{min}}(L)$, where $A_{\mathrm{min}}(L)$ is the minimal action of bounding disks. Categorical manifestations arise via the inequality

$\# Sh^{C,1}_\Lambda \leq \# Sh^{S,1}_\Lambda$

between the number of rank-1 objects in microsheaf categories localizing to the cylinder and to the "circle," precluding the possibility of squeezing when the inequality fails. Calculations for standard Lagrangians show $N_C > N_S$ below the critical radius, yielding sharp obstructions to embedding or isotoping long cylinders (Kilgore, 2024).

6. Summary Table: Key Rigidity Mechanisms for Long Cylinders

Mechanism Type	Mathematical Condition/Phenomenon	Reference
Capacity obstruction (length)	$$\ell(L) \geq \ln[c_k^-(\Lambda_-,\epsilon_-;\theta)] - \ln[c_k^+(\Lambda_+,\epsilon_+;\Psi_k(\theta))]$$	(Sabloff et al., 2015)
Barrier via Legendrian lift	No embedding $C_T(\Lambda)$ for $T > 1/k$ after removal of $\Lambda_k$	(Opshtein, 21 Dec 2025)
Legendrian non-squeezing	$\pi R^2 \geq A_{\min}(L)$ for prequantized cylinder embedding	(Kilgore, 2024)
Categorical (microsheaf)	$\# Sh^{C,1}_\Lambda \leq \# Sh^{S,1}_\Lambda$	(Kilgore, 2024)
Hamiltonian isotopy via invertible cobordism	$L_{\Lambda_0} \sim_{Ham} L_{\Lambda_1}$ even for non-diffeomorphic Legendrians	(Courte, 2015)

7. Applications and Further Directions

Long cylinders over Legendrians serve as foundational building blocks for the study of Lagrangian cobordisms, the analysis and computation of Legendrian invariants, and the explicit construction of SFT and TQFT structures. Their flexibility properties enable new constructions in high dimensions (notably flexible Lagrangian embeddings and the Mazur-trick Hamiltonian isotopies). Conversely, their rigidity—expressed through quantitative invariants such as capacities, action, symplectic area, and categorical counts—delimit the range of possible embeddings and isotopies. Prequantization bundle techniques and the use of Legendrian barriers provide explicit, geometric means of obstructing the existence of long cylinders in both high- and low-dimensional contact and symplectic manifolds.

A plausible implication is that future research will further refine the categorical and analytic obstructions to embedding long cylinders, and extend barrier and non-squeezing theorems to broader classes of Legendrians and contact manifolds. Interactions with microlocal sheaf theory, as well as the study of flexible versus rigid topological and categorical invariants, continue to drive the development of the field.