Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

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 ∈ (-∞, W̄₀ )∪ \ 0\ ∪ (W₀, +∞), where W̄₀ <0 and W₀ >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 ≥ 0, wherea it is a geodesic of nonvanishing curvature when W is a negative constant. Specially, in the first Heisenberg group 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 M(0) and provide a new proof of the fact that the total curvature of γ⊂ M(0) with strictly positive curvature equals $2π$.