Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

Abstract

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 $W\ge 0$, 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 $\gamma \subset \mathbb{M}(0)$ with strictly positive curvature equals $2\pi$.