C0 contact geometry of isotropic submanifolds

Homeomorphism is called contact if it can be written as C0-limit of contactomorphisms. The contact version of Eliashberg-Gromov rigidity theorem states that smooth contact homeomorphisms preserve contact structure. Submanifold L of a contact manifold (Y, ξ) is called isotropic if ξ|TL =0. Isotropic submanifolds of maximal dimension are called Legendrian, otherwise we call them subcritical isotropic.  In this talk, we will try to answer whether the isotropic property is preserved by contact homeomorphisms. It is expected that subcritical isotropic submanifolds are flexible, while we expect that Legendrians are rigid. We show that subcritical isotropic curves are flexible, and we give a new proof of the rigidity of Legendrians in dimension 3. Moreover, we provide a certain type of rigidity of Legendrians in higher dimensions.