Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow

TL;DR

This paper applies the distance comparison principle to the spatial curve shortening flow, establishing that for initial helix curves, the ratio of extrinsic to intrinsic distance is non-decreasing, and proves a Grayson type theorem for spatial curves.

Contribution

It extends the distance comparison principle and Grayson type theorem to three-dimensional spatial curves in the curve shortening flow.

Findings

The ratio of extrinsic to intrinsic distance is non-decreasing for initial helix curves.

A Grayson type theorem is proved for various spatial curves.

The results deepen understanding of spatial curve evolution under shortening flow.

Abstract

In this paper, we use the distance comparison principle, first been developed by G. Huisken, to study the spatial curve shortening flow. We have got the result that if the initial curve is the helix, then the local minimum of the ratio of the extrinsic and intrinsic distance is non-decreasing. And we have proved a Grayson type theorem for a variety of spatial curves.