# Morse flow trees and Legendrian contact homology in 1-jet spaces

@article{Ekholm2007MorseFT, title={Morse flow trees and Legendrian contact homology in 1-jet spaces}, author={Tobias Ekholm}, journal={Geometry \& Topology}, year={2007}, volume={11}, pages={1083-1224} }

Let L ⊂ J 1 (M) be a Legendrian submanifold of the 1-jet space of a Riemannian n-manifold M. A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid pseudo-holomorphic disks in TM, with boundary on the projection of L and asymptotic to the double points of this projection at punctures, provided n ≤ 2, or provided n > 2 and the front of L has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact…

## Figures from this paper

## 69 Citations

### A note on the infinite number of exact Lagrangian fillings for spherical spuns

- MathematicsPacific Journal of Mathematics
- 2022

In this short note we discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with infinite number of exact Lagrangian fillings up to Hamiltonian isotopy.…

### An exact sequence for Legendrian links

- Mathematics
- 2015

A result of Bourgeois, Ekholm and Eliashberg [4] describes the linearized contact homology of the boundary of a symplectic cobordism obtained by Legendrian surgery in terms of the cyclic homology of…

### Knot contact homology

- Mathematics
- 2013

The conormal lift of a link K in ℝ3 is a Legendrian submanifold ΛK in the unit cotangent bundle U∗ℝ3 of ℝ3 with contact structure equal to the kernel of the Liouville form. Knot contact homology, a…

### Functorial LCH for immersed Lagrangian cobordisms

- MathematicsJournal of Symplectic Geometry
- 2021

For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in…

### Cellular Legendrian contact homology for surfaces, part II

- Mathematics, BiologyInternational Journal of Mathematics
- 2019

It is proved that the Cellular DGA defined in [Cellular computation of Legendrian contact homology for surfaces, preprint (2016)] is stable tame isomorphic to the LegendrianContact homology DGA, modulo the explicit construction of a specific Legendrian surface.

### A Seifert-van Kampen Theorem for Legendrian Submanifolds and Exact Lagrangian Cobordisms

- Mathematics
- 2017

We prove a Seifert-van Kampen theorem for Legendrian submanifolds and exact Lagrangian cobordisms, and use it to calculate the change in the DGA caused by critical Legendrian ambient surgery.

### Generating families and augmentations for Legendrian surfaces

- Mathematics
- 2017

We study augmentations of a Legendrian surface $L$ in the $1$-jet space, $J^1M$, of a surface $M$. We introduce two types of algebraic/combinatorial structures related to the front projection of $L$…

### Orienting Moduli Spaces of Flow Trees for Symplectic Field Theory

- Mathematics
- 2016

This thesis consists of three scientific papers dealing with invariants of Legendrian and Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to contact and ...

### Orientations of Morse flow trees in Legendrian contact homology

- Mathematics
- 2015

Let L be a spin Legendrian submanifold of the 1-jet space of a smooth manifold. We prove that the Legendrian contact homology of L with integer coefficients can be computed using Morse flow trees. We…

### Invariants of Legendrian products

- Mathematics
- 2014

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter

## References

SHOWING 1-10 OF 23 REFERENCES

### The unregularized gradient flow of the symplectic action

- Mathematics
- 1988

The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this…

### Zero-loop open strings in the cotangent bundle and Morse homotopy

- Mathematics
- 1997

0. Introduction. Many important works in symplectic geometry and topology are regarded as the symplectization or the quantization of the corresponding results in ordinary geometry and topology. One…

### 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…

### NON-ISOTOPIC LEGENDRIAN SUBMANIFOLDS IN R

- Mathematics
- 2005

In the standard contact (2n+1)-space when n > 1, we construct infinite families of pairwise non-Legendrian isotopic, Legendrian n-spheres, n-tori and surfaces which are indistinguishable using…

### Some properties of holomorphic curves in almost complex manifolds

- Mathematics
- 1994

The study of holomorphic curves in almost complex manifolds can be viewed as the confluence of two fields.

### 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…

### Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions

- Mathematics
- 1992

We give a proof of the theorem of removing isolated singularities of pseudo-holomorphic curves with Lagrangian boundary conditions and bounded symplectic area. The proof is a combination of some…

### On Gradient Dynamical Systems

- Mathematics
- 1961

We consider in this paper a Co vector field X on a Co compact manifold Mn (&M, the boundary of M, may be empty or not) satisfying the following conditions: (1) At each singular point /8 of X, there…

### Monopoles on asymptotically flat manifolds

- Mathematics
- 1995

We consider the relation between the theory of monopoles on ℝ 3 and the analogous problem on a manifold M obtained from ℝ3 by forming the connected sum with a compact manifold M. The main result is…