Lagrangian Constant Cycle Subvarieties in Lagrangian Fibrations

We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold X whose general fiber is of dimension strictly between 0 and dim X is rationally connected. Using this result, we construct for any hyper-Kahler manifold X admitting a Lagrangian fibration a Lagrangian constant cycle subvariety Sigma(H) in X which depends on a divisor class H whose restriction to some smooth Lagrangian fiber is ample. If dim X = 4, we also show that up to a scalar multiple, the class of a zero-cycle supported on Sigma(H) in CH0(X) depend neither on H nor on the Lagrangian fibration (provided b(2)(X) >= 8).