Upper bounds for the Lagrangian cobordism relation on Legendrian links

Lagrangian cobordism induces a preorder on the set of Legendrian links in any contact 3-manifold.We show that any finite collection of null-homologous Legendrian links in a contact 3-manifold with a common rotation number has an upper bound with respect to the preorder.In particular, we construct an exact Lagrangian cobordism from each element of the collection to a common Legendrian link.This construction allows us to define a notion of minimal Lagrangian genus between any two null-homologous Legendrian links with a common rotation number.57K33; 53D12, 57K10