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Upper bounds for the Lagrangian cobordism relation on Legendrian links
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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 tight 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.
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Lagrangian concordance is not a partial order in high dimensions
In R^{4n+1} with n > 1, there exist pairs of non-isotopic loose Legendrian spheres with Lagrangian concordances in both directions, so Lagrangian concordance is not a partial order.
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