The Harnack inequality without convexity for curve shortening flow

TL;DR

The paper proves an alternative Harnack inequality for curve shortening flow that does not require convexity, using the evolution of swept area ${\mathcal{A}}$ and turning function ${\Psi}$ to form the Harnack quantity ${\mathcal{H}}={\mathcal{A}}-t{\Psi}$. By establishing a heat-equation structure for ${\mathcal{H}}$ on a fixed domain and controlling boundary behavior via turning-angle lemmas, it deduces sharp bounds on ${\mathcal{H}}$, yielding explicit graphicality time-scales for curves with radial ends. A key application shows that after a computable time depending on initial swept-area spread and sector angle, the evolving curve becomes a global graph with a gradient bound, revealing a delayed parabolic regularity phenomenon. The work also connects to Hamilton’s classical Harnack inequality: in the convex (radial-end) case, the new inequality implies the same pointwise curvature estimates, and, under polar graphical reformulations, recovers the familiar curvature decay $\kappa\le (1/(2t))D$, with equality for the $\beta$-wedge expander, illustrating consistency with the classical theory.

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.

Figures (14)

• Figure 1: How long before the evolution of this curve becomes graphical? Theorem \ref{['graphical_cor']} gives a precise time.
• Figure 2: Image of the initial data $\gamma(\cdot,0)$ of a curve shortening flow $\gamma$ with initially radial ends at angles $a=\frac{3\pi}{4}$ and $b=\frac{\pi}{4}$.
• Figure 3: The swept area ${\mathcal{A}}(v,w)$ is the blue shaded area minus the red dotted area.
• Figure 4: Illustration of a solution with ${\mathcal{A}}_-=0$. Each dotted radial line intersects the curve $\gamma(\cdot, t)$ precisely once.
• Figure 5: Qualitative picture of the $\beta$-wedge solution $\gamma_\beta$ at $t=1$.

Theorems & Definitions (49)

• Definition 1.1: Uniformly proper CSF
• Definition 1.2
• Definition 1.3
• Remark 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7: Alternative Harnack
• Theorem 1.8
• Theorem 1.9: Polar graphical Harnack inequality
• Remark 1.10