The Harnack inequality without convexity for curve shortening flow

TL;DR

This paper establishes a new Harnack inequality for curve shortening flow that does not require convexity, enabling analysis of more complex initial curves and their evolution over time.

Contribution

It introduces an alternative Harnack inequality for curve shortening flow without convexity assumptions, expanding applicability to arbitrary initial curves.

Findings

Provides explicit time bounds for curves to become graphical.

Establishes estimates for polar graphical flows approaching expanding solutions.

Links the new inequality to Hamilton's classical Harnack inequality in convex cases.

Abstract

In 1995, Hamilton introduced a Harnack inequality for convex solutions of the mean curvature flow. In this paper we prove an alternative Harnack inequality for curve shortening flow, i.e. one-dimensional mean curvature flow, that does not require any assumption of convexity. For an initial proper curve in the plane whose ends are radial lines but which is otherwise arbitrarily wild, we use the Harnack inequality to give an explicit time by which the curve shortening flow evolution must become graphical. This gives a new instance of delayed parabolic regularity. The Harnack inequality also gives estimates describing how a polar graphical flow with radial ends settles down to an expanding solution. Finally, we relate our Harnack inequality to Hamilton's by identifying a pointwise curvature estimate implied by both Harnack inequalities in the special case of convex flows.