Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908563529531392 |
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| author | Cui, Jingshi Zhao, Peibiao |
| author_facet | Cui, Jingshi Zhao, Peibiao |
| contents | In this paper, we introduce a kind of inverse mean curvature flow (1.2) in a Sasakian sub-Riemannian 3-manifold $M$ for Legendrian curves, which slightly differs from the classical one, and confirm that this flow preserves the Legendrian condition and increases the length of curves. We establish the long-time existence of the flow (1.2) when the Webster scalar curvature $W$ of $M$ satisfies $ W \in (-\infty, \bar{W}_{0} )\cup \{ 0\} \cup (W_{0}, +\infty)$, where $\bar{W}_{0} <0$ and $W_{0} >0$ are constants. Moreover, we derive that the local limit curve (the asymptotic behavior) along the flow (1.2) is a geodesic of vanishing curvature when $W \geq 0$, wherea it is a geodesic of nonvanishing curvature when $W$ is a negative constant.
Specially, in the first Heisenberg group $\mathbb{M}(0)$, we further construct a length-preserving flow (1.3) via a dilation of the flow (1.2) and show that closed Legendrian curves converge to Euclidean helices with vertical axis. By exploiting the properties of the flow (1.3), we establish a Minkowski-type formula for Legendrian curves in $\mathbb{M}(0)$ and provide a new proof of the fact that the total curvature of $γ\subset \mathbb{M}(0)$ with strictly positive curvature equals $2π$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_23865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds Cui, Jingshi Zhao, Peibiao Differential Geometry In this paper, we introduce a kind of inverse mean curvature flow (1.2) in a Sasakian sub-Riemannian 3-manifold $M$ for Legendrian curves, which slightly differs from the classical one, and confirm that this flow preserves the Legendrian condition and increases the length of curves. We establish the long-time existence of the flow (1.2) when the Webster scalar curvature $W$ of $M$ satisfies $ W \in (-\infty, \bar{W}_{0} )\cup \{ 0\} \cup (W_{0}, +\infty)$, where $\bar{W}_{0} <0$ and $W_{0} >0$ are constants. Moreover, we derive that the local limit curve (the asymptotic behavior) along the flow (1.2) is a geodesic of vanishing curvature when $W \geq 0$, wherea it is a geodesic of nonvanishing curvature when $W$ is a negative constant. Specially, in the first Heisenberg group $\mathbb{M}(0)$, we further construct a length-preserving flow (1.3) via a dilation of the flow (1.2) and show that closed Legendrian curves converge to Euclidean helices with vertical axis. By exploiting the properties of the flow (1.3), we establish a Minkowski-type formula for Legendrian curves in $\mathbb{M}(0)$ and provide a new proof of the fact that the total curvature of $γ\subset \mathbb{M}(0)$ with strictly positive curvature equals $2π$. |
| title | Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.23865 |