The volume-preserving Willmore flow

Abstract

We consider a closed surface in ${\mathbb{R}}^3$ evolving by the volume-preserving Willmore flow and prove a lower bound for the existence time of smooth solutions. For spherical initial surfaces with Willmore energy below $8\pi$ we show long time existence and convergence to a round sphere by performing a suitable blow-up and by proving a constrained Lojasiewicz-Simon inequality.

38 pages. Shortened and restructured final version. To appear in Nonlinear Analysis

References in corpus (5)

• Surface diffusion flow near spheres
• Lifespan theorem for constrained surface diffusion flows
• The Willmore Flow of Tori of Revolution
• Global existence and full convergence of the Möbius-invariant Willmore flow in the $3$-sphere
• A Reverse Isoperimetric Inequality and its Application to the Gradient Flow of the Helfrich Functional

Cited by in corpus (5)

• The Willmore flow with prescribed isoperimetric ratio
• Existence and convergence of the length-preserving elastic flow of clamped curves
• A dynamic approach to heterogeneous elastic wires
• Willmore Flow of Complete Surfaces
• Łojasiewicz-Simon inequalities for minimal networks: stability and convergence