Seminar on Geometric Analysis and Its Applications: The curve shortening flow and Gage’s area-preserving flow of star-shaped curves

Speaker: Laiyuan Gao (Jiangsu Normal University)

Time: 2025 Jun 19 10:00-11:30am
Location: R1310, SIMIS
Zoom Meeting ID: 880 8881 7737 (Passcode: 042893)

Abstract:
Mayer asks a question what closed embedded and nonconvex initial curves enjoy global existence under Gage’s area-preserving flow. A folklore conjecture since 2012 says that GAPF evolves smooth, embedded and star-shaped initial curves globally. In this talk, 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.