Work

Curve shortening flow with an ambient force field

Year: 2025

Type: article

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 $$L^{\infty }$$ L ∞ 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.

Source: Calculus of Variations and Partial Differential Equations

DOI: 10.1007/s00526-025-02983-x

Authors: Samuel Cuthbertson, Glen Wheeler, Valentina-Mira Wheeler

Institution: University of Wollongong

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funders: University of Wollongong, Australian Research Council