Curve Shortening Flow of Space Curves with Convex Projections

2024/10/10 by Qi Sun, Sun, Qi · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #advanced mathematical theories #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2410.08399

Abstract

We show that under Space Curve Shortening flow any closed immersed curve in \mathbb Rⁿ whose projection onto ℝ²×\0\ is convex remains smooth until it shrinks to a point. Throughout its evolution, the projection of the curve onto ℝ²×\0\ remains convex. As an application, we show that any closed immersed curve in \mathbb Rⁿ can be perturbed to an immersed curve in \mathbb Rⁿ⁺² whose evolution by Space Curve Shortening shrinks to a point.

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