The volume-preserving Willmore flow

TL;DR

This paper studies the volume-preserving Willmore flow for closed surfaces in three-dimensional space, establishing conditions for long-term smooth evolution and convergence to a sphere, especially for spherical initial surfaces with low Willmore energy.

Contribution

It proves a lower bound on the existence time for smooth solutions and demonstrates long-term existence and convergence for certain initial conditions using advanced analytical techniques.

Findings

Long-term existence for spherical surfaces with Willmore energy below 8π.

Convergence to a round sphere under specified conditions.

Application of a constrained Lojasiewicz-Simon inequality.

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.