On the mean curvature flow and Hamiltonians diffeomorphisms

Mean curvature flow plays an important role in geometric, mostly while searching for volume minimising submanifold. As a very powerful analysis technics, it is an interplay tool between analysis and geometry. It appears specially helpful to look for a map between manifold with somehow nice properties. It gives rise to a so called mean curvature flow method which helps and is widely used in the study of isotopy type of diffeomorphisms groups leading to some results in symplectic topology.