Harnack inequalities for the curve shortening flow

TL;DR

This paper presents an algorithmic method to derive Harnack inequalities for the curve shortening flow, successfully reproducing Hamilton's inequality through a systematic procedure.

Contribution

It introduces a novel algorithmic approach for deriving Harnack quantities, providing a new systematic way to obtain classical inequalities in geometric flows.

Findings

Derived Hamilton's Harnack inequality for curve shortening flow

Proposed an algorithmic framework for Harnack quantity determination

Validated the approach by reproducing known inequalities

Abstract

We offer an algorithmic approach for determining Harnack quantities for the curve shortening flow and we show how, following this procedure, one can obtain Hamilton's Harnack inequality for this flow $\kappa_t + \frac{1}{2t}\kappa \geq \frac{\kappa_s^2}{\kappa}$, where $\kappa$ is the curvature of the curve being deformed by the flow.