Continuous dependence of curvature flow on initial conditions

2021/06/16 by Dobbins, Michael Gene
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2106.08907

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 Fréchet 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.

Related

• The continuity of curvature flow in Fréchet distance
• The Dirichlet problem for curve shortening flow
• The evolution of Jordan curves on \mathbbS² by curve shortening flow
• Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere
• Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows