The Willmore flow with prescribed isoperimetric ratio

Abstract

We introduce a non-local $L^2$-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.

arXiv commentFinal version. 30 pages, 1 figure. To appear in Comm. Partial Differential Equations

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