Star-shaped Curves under Gage's Area-preserving Flow and the CSF

TL;DR

This paper proves that Gage's area-preserving flow preserves the global existence of smooth, embedded, star-shaped curves, confirming a longstanding conjecture, while also providing a counterexample showing star-shapedness may not be preserved.

Contribution

It confirms the folklore conjecture that star-shaped initial curves evolve globally under GAPF and introduces a counterexample for star-shapedness preservation in CSF.

Findings

GAPF preserves global existence for star-shaped curves

Constructed a counterexample showing star-shapedness may not be preserved in GAPF

Provided a negative answer to star-shapedness preservation in CSF

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.