A note on expansion of convex plane curves via inverse curvature flow

TL;DR

This paper extends the use of a comparison function to analyze the inverse curvature flow of convex plane curves, proving that such flows exist for all time and become circular, similar to the shrinking case.

Contribution

It provides a new proof that convex curves under inverse curvature flow exist globally and converge to a circle, using a comparison function originally applied to curve shortening flow.

Findings

Flow exists for all times

Curves become round and converge to a circle

Comparison function effectively estimates curvature decay

Abstract

Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this comparison function they estimate the length of any chord from below in terms of the arc length between its endpoints and elapsed time. They apply this estimate to short segments and deduce directly that the maximum curvature decays exponentially to the curvature of a circle with the same length. We consider the expansion of convex curves under inverse (mean) curvature flow and show that the above comparison function also works in this case to obtain a new proof of the fact that the flow exists for all times and becomes round in shape, i.e. converges smoothly to the unit circle after an appropriate rescaling.