Morse flow trees and Legendrian contact homology in 1–jet spaces

Abstract

Let $L\subset J^1\left(M\right)$ 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 $T^{\ast }M$, with boundary on the projection of $L$ and asymptotic to the double points of this projection at punctures, provided $n\le 2$, 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 homology of $L$ in terms of Morse theory.