  Recently, there are a great deal of work done which connects the Legendrian
isotopic problem with contact invariants. The isotopic problem of Legendre
curve in a contact 3-manifold was studies via the Legendrian curve shortening
flow which was introduced and studied by K. Smoczyk. On the other hand, in the
SYZ Conjecture, one can model a special Lagrangian singularity locally as the
special Lagrangian cones in C^{3}. This can be characterized by its link which
is a minimal Legendrian surface in the 5-sphere. Then in these points of view,
in this paper we will focus on the existence of the long-time solution and
asymptotic convergence along the Legendrian mean curvature flow in higher
dimensional {\eta}-Einstein Sasakian (2n+1)-manifolds under the suitable
stability condition due to the Thomas-Yau conjecture.
