We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair consisting of an exact symplectic manifold and an exact Lagrangian cobordism which agrees with cylinders over Legendrian links and at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of to that of . We give a gradient flow tree description of the DGA maps for certain pairs , which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.
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Tobias Ekholm, Ko Honda, Tamás Kálmán, Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. 18 (2016), no. 11, pp. 2627–2689DOI 10.4171/JEMS/650