The Willmore flow with prescribed isoperimetric ratio

Abstract

We introduce a non-local L²-gradient flow for the Willmore energy of immersed surfaces which preserves the isoperimetric ratio. For spherical initial data with energy below an explicit threshold, we show long-time existence and convergence to a Helfrich immersion. This is in sharp contrast to the locally constrained flow, where finite time singularities occur.

Keywords:

• Willmore flow
• Helfrich energy
• isoperimetric ratio
• Łojasiewicz–Simon inequality
• non-local geometric evolution equation

2020 Mathematics Subject Classification:

• Primary: 53E40
• Secondary: 35B40, 35K41

Acknowledgments

The author would like to thank Anna Dall’Acqua for helpful discussions and comments. In addition, the author is grateful to the referees for their careful reading and their valuable comments on the original manuscript.

Disclosure statement

The author reports there are no competing interests to declare.

Additional information

Funding

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant 404870139 and the Austrian Science Fund (FWF) under Grant P 32788-N.