2023/12/26 by Chang, Shu-Cheng, Wu, Chin-Tung, Zhang, Liuyang
#53C44 #53C56 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.15996
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 \mathbbS2× \mathbbS3. By using the monotonicity formula and blow-up analysis, we obtain the minimal Legendrian graph in \mathbbS2× \mathbbS3. Finally, we will address the rigidity theorem of 2-dimensional Legendrian self-shrinkers in ℝ5. We are able to reconstruct the Harvey-Lawson special Lagrangian cone in ℂ3 from this Legendrian self-shrinker. The partial classification is also provided if the squared norm of the second fundamental form is constant.