This paper focuses on a length-preserving flow for star-shaped curves with respect to the origin. Under the length-preserving flow, the evolving curve keeps star-shapedness and converges smoothly to a circle, which can be regarded as a Grayson-type theorem for star-shaped curves under this flow.

中文翻译：

---

星形曲线的格雷森型定理

本文重点研究星形曲线相对于原点的长度保持流。在保长流下，演化曲线保持星形并平滑地收敛到圆，这可以看作是该流下星形曲线的格雷森型定理。