Minimal surface generating flow for space curves of non-vanishing torsion

Abstract

<p style='text-indent:20px;'>This article introduces new geometric flow for space curves with positive curvature and torsion. Curves evolving according to this motion law trace out a zero mean curvature surface. First, the geometry of surfaces defined as trajectories of moving curves is analyzed and the minimal surface generating flow is derived. Then, an upper bound for the area of the generated minimal surface and for the terminating time of the flow is provided.</p>

Geometry of solutions of the geometric curve flows in space

“…(2.4) Definition 2.1. [21,26] (Trajectory surface) For given velocities v T , v N and v B , an initial curve Γ 0 and terminal time t max , we define the trajectory surface ∑ t max as ∑ t max := t∈[0,t max ) Γ t .…”

Geometry of solutions of the geometric curve flows in space

“…) Definition 1. 21,26 (Trajectory surface) For given velocities 𝑣 𝑇 , 𝑣 𝑁 and 𝑣 𝐵 , an initial curve Γ 0 and terminal time 𝑡 𝑚𝑎𝑥 , we define the trajectory surface ∑ 𝑡 𝑚𝑎𝑥 as…”

“…Trajectory surfaces have been studied in 1,21,23,26 for the following cases, such as the binormal flow (i.e., 𝑣 𝑇 = 𝑣 𝑁 = 0, 𝑣 𝐵 = 𝜅), the curve shortening flow (i.e., 𝑣 𝑇 = 𝑣 𝐵 = 0, 𝑣 𝑁 = 𝜅), minimal surface generating flow (i.e., 𝑣 𝑇 = 𝑣 𝐵 = 0, 𝑣 𝑁 = 𝜏 − 1 2 ), inextensible flows (i.e., geometric flows with 𝜕𝑔 𝜕𝑡 = 0).…”

“…(iii) A minimal trajectory surface ∑ 3 is the surface traced out by a family of curves {Γ 𝑡 } 𝑡∈[0,𝑡 𝑚𝑎𝑥 ) in ℝ 3 as it evolves over time according to the minimal surface generating flow (see 26 ):…”

“…Trajectory surfaces have been studied for the special case of inextensible flows in 21 , curves flows of elastic rods in 22 , and the curve shortening flow in 23,24,25 . Recently, J. Minarčík and M. Beneš 26 introduced minimal surface generating flow for space curves of non-vanishing torsion in Euclidean 3-space ℝ 3 , and analyzed some properties of space curves evolved by the minimal surface generating flow.…”