The Smale Conjecture and Minimal Legendrian Graph in $\mathbb{S}^{2}\times \mathbb{S}^{3}$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915442502664192 |
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| author | Chang, Shu-Cheng Wu, Chin-Tung Zhang, Liuyang |
| author_facet | Chang, Shu-Cheng Wu, Chin-Tung Zhang, Liuyang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15996 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Smale Conjecture and Minimal Legendrian Graph in $\mathbb{S}^{2}\times \mathbb{S}^{3}$ Chang, Shu-Cheng Wu, Chin-Tung Zhang, Liuyang Differential Geometry 53C44, 53C56 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. |
| title | The Smale Conjecture and Minimal Legendrian Graph in $\mathbb{S}^{2}\times \mathbb{S}^{3}$ |
| topic | Differential Geometry 53C44, 53C56 |
| url | https://arxiv.org/abs/2312.15996 |