The unregularized gradient flow of the symplectic action

Abstract The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P . To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy‐Riemann equation with respect to a suitable almost complex structure on P . In particular, we prove compactness and transversality results for the set of bounded trajectories.