The Parametric Willmore Flow

TL;DR

This paper proves a positive minimal existence time for the parametric Willmore flow starting from smooth initial data, and discusses extending the flow to weaker Lipschitz initial immersions based on geometric data and conservation laws.

Contribution

It establishes a minimal existence time for the flow for smooth data and explores the potential for defining the flow for weak Lipschitz initial immersions.

Findings

Positive minimal existence time for smooth initial data

Dependence of existence time solely on geometric data

Potential extension of the flow to weak Lipschitz immersions

Abstract

We establish a minimal positive existence time of the parametric Willmore flow for any smooth initial data (smooth immersion of a closed oriented surface). The minimal existence time is a function exclusively of geometric data which in particular are all well defined for general weak lipschitz $W^{2,2}$ immersions. This fact combined with the conservation law formulation of the equation given by the first author in 2008 opens the possibility for defining the Willmore flow for weak lipschitz $W^{2,2}$ initial data.