Curve flows with a global forcing term

TL;DR

This paper investigates the evolution of embedded plane curves under curve shortening flow with a global forcing term, establishing conditions for singularity prevention and convergence to circles.

Contribution

It introduces a sharp distance comparison principle for such flows, proves finite-time singularity exclusion for bounded forcing, and shows convergence to circles for certain closed curves.

Findings

Distance comparison principle analogous to Huisken's established.

Finite-time singularities are excluded for flows with bounded forcing.

Closed curves with non-vanishing enclosed area converge to circles.

Abstract

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. Firstly, we prove an analogue to Huisken's distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below $-\pi$, and show that this condition is sharp. Secondly, for bounded forcing terms, we exclude singularities in finite time. Thirdly, for immortal flows of closed curves whose forcing terms provide non-vanishing enclosed area and bounded length, we show convexity in finite time and smooth and exponential convergence to a circle.