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

The Smale Conjecture and Minimal Legendrian Graph in $\mathbb{S}^{2}\times \mathbb{S}^{3}$