Curve Shortening Flow of Space Curves with Convex Projections

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.