Curve Shortening Flow of Space Curves with Convex Projections

TL;DR

This paper proves that space curves with convex projections under curve shortening flow remain smooth and shrink to a point, and demonstrates how to perturb curves to achieve this behavior in higher dimensions.

Contribution

It establishes the preservation of convex projections and smoothness for space curves under flow, and introduces a method to perturb curves for controlled shrinking in higher dimensions.

Findings

Convex projections are preserved during flow.

Curves remain smooth until shrinking to a point.

Perturbation method for higher-dimensional curves.

Abstract

We show that under Space Curve Shortening flow any closed immersed curve in $\mathbb R^n$ whose projection onto $\mathbb{R}^2\times\{\vec{0}\}$ is convex remains smooth until it shrinks to a point. Throughout its evolution, the projection of the curve onto $\mathbb{R}^2\times\{\vec{0}\}$ remains convex. As an application, we show that any closed immersed curve in $\mathbb R^n$ can be perturbed to an immersed curve in $\mathbb R^{n+2}$ whose evolution by Space Curve Shortening shrinks to a point.