A comparison theorem for the isoperimetric profile under curve-shortening flow

We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve-shortening flow: if the isoperimetric profile of the region enclosed by the initial curve is greater than that of some "model" convex region with exactly four vertices and with reflection symmetry in both axes, then the inequality remains true for the isoperimetric profiles of the evolved regions. We apply this using the "paperclip" solution as the model region to deduce sharp time-dependent upper bounds on curvature for arbitrary embedded closed curves evolving by the normalized curve-shortening flow. A slightly different comparison also gives lower bounds on curvature, and the result is a simple and direct proof of Grayson's theorem without use of any blowup or compactness arguments, Harnack estimates or classification of self-similar solutions.