Long-term behavior of curve shortening flow in $\mathbb{R}^3$

TL;DR

This paper investigates the long-term behavior of space curves under the curve shortening flow in three-dimensional space, revealing differences from planar curves and establishing new properties and principles specific to space curves.

Contribution

It demonstrates that convexity is not preserved for space curves but their projections remain convex, and it generalizes the Avoidance principle to spherical curves in 3D.

Findings

Convexity of space curves is not preserved during flow.

Orthogonal projections of space curves remain convex.

Avoidance principle is valid for spherical curves in $\

Abstract

Space curve motion describes dynamics of material defects or interfaces, can be found in image processing or vortex dynamics. This article analyses some properties of space curves evolved by the curve shortening flow. In contrast to the classical case of shrinking planar curves, space curves do not obey the Avoidance principle in general. They can lose their convexity or develop non-circular singularities even if they are simple. In the first part of the text, we show that even though the convexity of space curves is not preserved during the motion, their orthogonal projections remain convex. In the second part, the Avoidance principle for spherical curves under the curve shortening flow in $\mathbb{R}^3$ is shown by generalizing the arguments developed by Hamilton and Gage.