  Let ({\Sigma}, {\omega}) be a compact Riemann surface with constant curvature
c. In this work, we proved that the mean curvature flow of a given Hamiltonian
diffeomorphism on {\Sigma} provides a smooth path in Ham({\Sigma}), the group
of all Hamiltonian diffeomorphisms of {\Sigma}. This result gives a proof, in
the case of graph of Hamiltonian diffeomorphisms to the conjecture of Thomas
and Yau asserting that the mean curvature flow of a compact embedded Lagrangian
submanifold S with zero Maslov class in a Calabi- Yau manifolds M exists for
all time and converges smoothly to a special Lagrangian submanifold in the
Hamiltonian isotopy class of S.
