Submanifold averaging in riemannian and symplecitc geometry

We give a construction to obtain canonically an isotropic average'' of given $C^1$-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem (obtained in collaboration with H. Karcher) and apply Moser’s trick’'. We also present an application to Hamiltonian group actions.