Continuous dependence of curvature flow on initial conditions

TL;DR

This paper proves that the evolution of a Jordan curve on the sphere under curvature flow depends continuously on the initial curve in the Fréchet distance, even for non-smooth initial curves and as time approaches infinity.

Contribution

It establishes continuous dependence of curvature flow solutions on initial conditions for non-smooth curves on the sphere, extending prior results to the sphere setting.

Findings

Curve evolution depends continuously on initial data in Fréchet distance.

Continuity holds even as time approaches infinity.

Results extend to non-smooth initial curves on the sphere.

Abstract

We study the evolution of a Jordan curve on the 2-sphere by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the evolution of the curve depends continuously on the initial curve in Fr\'echet distance in the case where the curve bisects the sphere. This even holds in the limit as time goes to infinity. This builds on Joseph Lauer's work on existence and uniqueness of solutions to the curvature flow problem on the sphere when the initial curve is not smooth.