The blow up analysis of the general curve shortening flow

TL;DR

This paper analyzes the behavior of curves under a general flow that shortens them, deriving a nonlinear evolution equation for curvature and describing how curves contract to a point over finite time.

Contribution

It introduces a detailed asymptotic analysis of the curvature evolution under the general curve shortening flow, extending previous results to a broader class of flows.

Findings

Curvature satisfies a nonlinear evolution equation.

Closed curves contract to a point in finite time.

Asymptotic behavior of curves near singularity is characterized.

Abstract

It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.