Curve shortening-straightening flow for non-closed planar curves with infinite length

TL;DR

This paper studies a geometric flow called the shortening-straightening flow for non-closed planar curves with infinite length, proving long-term existence, convergence to stationary solutions, and classifying these solutions.

Contribution

It establishes the long-time existence and convergence of the flow for infinite-length curves and classifies all stationary solutions, advancing understanding of geometric flows for non-closed curves.

Findings

Proved long-time existence of the flow.

Showed convergence to stationary solutions.

Classified stationary solutions.

Abstract

We consider a motion of non-closed planar curves with infinite length. The motion is governed by a steepest descent flow for the geometric functional which consists of the sum of the length functional and the total squared curvature. We call the flow shortening-straightening flow. In this paper, first we prove a long time existence result for the shortening-straightening flow for non-closed planar curves with infinite length. Then we show that the solution converges to a stationary solution as time goes to infinity. Moreover we give a classification of the stationary solution.