Lagrangian fillings and complicated Legendrian unknots

Sylvain Courte (1) , Tobias Ekholm (2)

1 IF - Institut Fourier
2 Department of Mathematics [Uppsala]

An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $\Sigma$ can be glued to itself along $\Sigma$. This gives a Legendrian embedding $\Lambda(L,L)$ of the double of $L$ into contact $(2n+1)$-space. We show that the Legendrian isotopy class of $\Lambda(L,L)$ is determined by formal data: the manifold $L$ together with a trivialization of its complexified tangent bundle. In particular, if $L$ is a disk then $\Lambda(L,L)$ is the Legendrian unknot.

Domaines: Géométrie symplectique [math.SG]

Sylvain Courte, Tobias Ekholm. Lagrangian fillings and complicated Legendrian unknots. Proceedings of the Gökova Geometry-Topology conference, 2017. ⟨hal-02009495⟩