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Curve Shortening Flow of Space Curves with Convex Projections

Qi Sun

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.

Curve Shortening Flow of Space Curves with Convex Projections

Abstract

We show that under Space Curve Shortening flow any closed immersed curve in whose projection onto is convex remains smooth until it shrinks to a point. Throughout its evolution, the projection of the curve onto remains convex. As an application, we show that any closed immersed curve in can be perturbed to an immersed curve in whose evolution by Space Curve Shortening shrinks to a point.

Paper Structure

This paper contains 7 sections, 49 theorems, 128 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Convexity of the projection
  3. Uniform convexity of the projection
  4. The three-point condition
  5. Proof of Theorem \ref{['main theorem']}(b) assuming the projection shrinks to a point
  6. Some facts about the Space Curve Shortening flow
  7. Proof of Proposition \ref{['long time behaviour under three point condition']}

Key Result

Theorem 1.5

Consider an embedded smooth curve $\gamma_0$ in $\mathbb{R}^n(n\geq 2)$, for which $P_{xy}|_{\gamma_0}$ is injective and the projection curve $\Bar{\gamma}_0$ is convex. Let $\gamma:S^1\times \left[0,T\right)\rightarrow\mathbb{R}^n$ be the solution to the space CSF with $\gamma(u,0)=\gamma_0(u)$. Th

Figures (5)

  • Figure 1: A planar figure eight and its perturbation
  • Figure 2: Snapshots of the evolution of a perturbation of the planar figure eight curve from different angles
  • Figure 3: An illustration of Lemma \ref{['perturb curves with one direction exactly two critical points']}. On the left we have the unperturbed curve in $\mathbb R^2$, and on the right is the perturbation in $\mathbb{R}^3$. Its projection, drawn in red, is an ellipse, hence convex. By Theorem \ref{['main theorem']}, the perturbed curve remains smooth until it shrinks to a point.
  • Figure 4: The left figure shows the curve $\gamma^{\epsilon}_0$ when $\epsilon=0$ and the right figure depicts the curve when $\epsilon$ is small and positive. The red curves are the corresponding projection curves in the $xy$-plane.
  • Figure 5: The upper branch

Theorems & Definitions (117)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 1.6
  • Lemma 1.7
  • Lemma 1.8
  • proof : Proof of Lemma \ref{['perturb curves with one direction exactly two critical points']}
  • Lemma 2.1: Lemma 1.4 of AltschulerGrayson
  • ...and 107 more