Legendrian Mean Curvature Flow in \(\eta \)-Einstein Sasakian Manifolds

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 studied 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 \({\mathbb {C}}^{3}\) ${\mathbb{C}}^3$ . This can be characterized by its link which is a minimal Legendrian surface in the 5-sphere. Then in these points of view, we will focus on the existence of the long-time solution and asymptotic convergnce along the Legendrian mean curvature flow in the \((2n+1)\) $(2n+1)$ -dimensional \(\eta \) $\eta$ -Einstein Sasakian manifolds under the suitable stability condition due to the Thomas-Yau conjecture.