Star-shaped Curves under Gage's Area-preserving Flow and the CSF
Differential Geometry
2024-12-25 v1
Abstract
Mayer asks a question what closed, embedded and nonconvex initial curves guarantee that Gage's area-preserving flow (GAPF) exists globally. A folklore conjecture since 2012 says that GAPF evolves smooth, embedded and star-shaped initial curves globally. In this paper, we prove this conjecture by using Dittberner's singularity analysis theory. A star-shaped ``flying wing" curve is constructed to show that GAPF may not always preserve the star-shapedness of evolving curves. This example is also a negative answer to Mantegazza's open problem whether the curve shortening flow (CSF) always preserves the star shape of the evolving curves.
Cite
@article{arxiv.2412.18102,
title = {Star-shaped Curves under Gage's Area-preserving Flow and the CSF},
author = {Laiyuan Gao and Shicheng Zhang and Yuntao Zhang},
journal= {arXiv preprint arXiv:2412.18102},
year = {2024}
}
Comments
17pages, 4 gigures