Curve Shortening Flow of Space Curves with Convex Projections

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.