# A note on exact Lagrangian cobordisms with disconnected Legendrian ends

@article{Chantraine2013ANO, title={A note on exact Lagrangian cobordisms with disconnected Legendrian ends}, author={Baptiste Chantraine}, journal={arXiv: Symplectic Geometry}, year={2013}, pages={1325-1331} }

We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian cobordism on which all primitive of the Liouville form is not constant on the negative end and such that the positive end is a stabilisation whereas the negative end admits augmentations. These examples emphasise point (i) of the definition of exact Lagrangian…

## 18 Citations

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## References

SHOWING 1-10 OF 13 REFERENCES

### Legendrian knots and exact Lagrangian cobordisms

- Mathematics
- 2012

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact…

### Obstructions to Lagrangian Cobordisms between Legendrians via Generating Families

- Mathematics
- 2011

The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact,…

### Lagrangian concordance of Legendrian knots

- Mathematics
- 2006

In this article, we define the notion of a Lagrangian concordance between two Legendrian knots analogous to smooth concordance in the Legendrian context. We show that Legendrian isotopic Legendrian…

### Some non‐collarable slices of Lagrangian surfaces

- Mathematics
- 2012

In this note, we define the notion of collarable slices of Lagrangian submanifolds. These are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder…

### Legendrian Contact Homology

- Mathematics

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P × R where P is an exact symplectic manifold is established. The class of such contact…

### Legendrian contact homology in $P \times \mathbb{R}$

- Mathematics
- 2007

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P x R, where P is an exact symplectic manifold, is established. The class of such contact…

### Legendrian Contact Homology in P X R

- Mathematics
- 2005

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic manifold is established. The class of such…

### The contact homology of Legendrian submanifolds in R2n+1

- Mathematics
- 2005

We define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. This…

### Differential algebra of Legendrian links

- Mathematics
- 2002

Let the space R = {(q, p, u)} be equipped with the standard contact form α = du− pdq. A link L ⊂ R3 is called Legendrian if the restriction of α to L vanishes. Two Legendrian links are said to be…

### Pseudo holomorphic curves in symplectic manifolds

- Mathematics
- 1985

Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called…