Stability conditions and Lagrangian cobordisms

In this paper we study the interplay between Lagrangian cobordisms and stability conditions. We show that any stability condition on the derived Fukaya category DFuk(M) of a symplectic manifold (M, omega) induces a stability condition on the derived Fukaya category of Lagrangian cobordisms DFuk(C x M). In addition, using stability conditions, we provide general conditions un- der which the homomorphism Theta : Omega(Lag) (M) -> K-0 (DFuk(M)), introduced by Biran and Cornea [6, 7], is an isomorphism. This yields a better understanding of how stability conditions affect Theta and it allows us to elucidate Haug's result, that the Lagrangian cobordism group of T-2 is isomorphic to K-0(DFuk(T-2)) [23].