  In this article, we recapture the Smale conjecture on a Sasakian $3$-sphere
via the Legendrian mean curvature flow. More precisely,~we deform the
area-preserving contactomorphism (symplectomorphism) of Sasakian $3$-spheres to
an isometry via the Legendrian mean curvature flow on the Legendrian graph in
$\mathbb{S}^{2}\times \mathbb{S}^{3}$. By using the monotonicity formula and
blow-up analysis, we obtain the minimal Legendrian graph in
$\mathbb{S}^{2}\times \mathbb{S}^{3}$. Finally, we will address the rigidity
theorem of $2$-dimensional Legendrian self-shrinkers in $\mathbb{R}^{5}$. We
are able to reconstruct the Harvey-Lawson special Lagrangian cone in
$\mathbb{C}^{3}$ from this Legendrian self-shrinker. The partial classification
is also provided if the squared norm of the second fundamental form is
constant.
