Using an averaging procedure by Alan Weinstein and ``Moser's
trick'' we will give a construction to obtain canonically an
``isotropic average`` of given $C^1$ close conpact isotropic submanifolds of a K\"ahler manifold. The isotropic average will be $C^0$ close to the given submanifolds, and the construction will be
equivariant with respect to K\"ahler diffeomorphisms.\\As a corollary we obtain that if an isotropic submanifold is almost invariant under an action by K\"ahler diffeomorphisms, then nearby there will be an invariant one.
Back to Geometry of Lagrangian Submanifolds