The Willmore flow with prescribed isoperimetric ratio

TL;DR

This paper introduces a non-local gradient flow for the Willmore energy that preserves the isoperimetric ratio, demonstrating long-term existence and convergence for certain initial conditions, unlike local flows which develop singularities.

Contribution

It presents a novel non-local flow for the Willmore energy that maintains the isoperimetric ratio and ensures long-time smooth evolution under specific conditions.

Findings

Flow preserves isoperimetric ratio during evolution

Long-time existence and convergence for initial data below energy threshold

Contrasts with local flows that develop finite time singularities

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.