# Length-constrained curve diffusion

## TL;DR

This paper proves that closed curves close to a circle evolve under length-constrained curve diffusion to become perfect circles exponentially fast, with no other closed translating or rotating solutions besides circles.

## Contribution

It establishes convergence to circles for length-constrained curve diffusion and rules out non-circular translating and rotating solutions.

## Key findings

- Curves near a circle converge exponentially to a round circle.
- No closed translating solutions exist for the flow.
- The only closed rotators are circles.

## Abstract

We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow and that the only closed rotators are circles.