Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow

TL;DR

This paper introduces a curvature flow with a nonlocal term that deforms one locally convex curve into another with the same winding number, preserving convexity and elastic energy, and converging to the target curve over time.

Contribution

It constructs a novel curvature flow that guarantees global existence, preserves key geometric properties, and achieves deformation into a target curve with the same elastic energy.

Findings

Flow exists globally and preserves local convexity.

Flow deforms initial curve into target curve if elastic energies match.

Curve deformation converges to the target as time approaches infinity.

Abstract

Let $X_0, \widetilde{X}$ be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve $X_0$ into $\widetilde{X}$. It is proved that this flow exits globally, preserves both the local convexity and the elastic energy of the evolving curve. If the two curves have same elastic energy then the curvature flow deforms the evolving curve into the target curve $\widetilde{X}$ as time tends to infinity.