Blow-up rates for the general curve shortening flow

TL;DR

This paper investigates how derivatives of curvature blow up as closed curves contract to a point under the general curve shortening flow, extending previous results on mean curvature flow.

Contribution

It generalizes a theorem by Gage and Hamilton to a broader class of curve shortening flows, providing new blow-up rate estimates for derivatives of curvature.

Findings

Derived explicit blow-up rates for curvature derivatives

Extended Gage and Hamilton's theorem to general flow cases

Provided mathematical framework for analyzing finite-time singularities

Abstract

The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.