Given a curve in the plane, let it deform in the direction of its curvature vector. It was conjectured that the curve would collapse in finite time to a point and that, as the curve collapsed, it would become asymptotically close to a shrinking standard circle. The authors prove this conjecture for a convex plane curve. In a more recent paper [J. Differ. Geom. 26, 285-314 (1987)], M. A. Grayson shows that any embedded curve eventually becomes convex, completely settling the conjecture.
Integral estimates (and the isoperimetric inequality) are used to obtain the necessary bounds for proving long time existence and convergence towards a shrinking circle. This is in contrast to G. Huisken’s work on deforming higher dimensional hypersurfaces by the mean curvature vector, where the Codazzi-Mainardi equations and the maximum principle are used.