Curve shortening flow with an ambient force field

TL;DR

This paper studies the anisotropic curve shortening flow with an ambient force, proving conditions under which curves shrink to round points or become non-convex, extending classical results to include ambient forces.

Contribution

It extends classical curve shortening flow results by incorporating ambient force fields and identifies conditions affecting convexity and shape evolution.

Findings

Closed embedded curves with large enough curvature shrink to round points.

Ambient force fields can prevent curves from remaining convex, leading to non-convexity.

Conditions are provided under which the classical shrinking behavior is preserved or altered.

Abstract

In this paper we consider the anisotropic curve shortening flow in the plane in the presence of an ambient force. We consider force fields in which all their derivatives are bounded in the $L^{\infty}$ sense. We prove that closed embedded curves that have a minimum of curvature sufficiently large shrink to round points. The method of proof follows along the same lines of Gage and Hamilton, in that we study a rescaling to prove curvature bounds. We additionally show that the influence of an ambient force field may make such a result untrue, by giving sufficient conditions on the ambient field that ensures eventual non-convexity of an initially convex curve evolving under the flow.