  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.
