The Parametric Approach to the Willmore Flow

TL;DR

This paper develops a parametric framework for Willmore gradient flows, allowing analysis of weak solutions, energy quantization, and singularities, with initial results on existence and uniqueness in a small-energy setting.

Contribution

It introduces a novel parametric approach to Willmore flow, enabling the study of weak solutions and singularities, which was not possible with previous methods.

Findings

Existence and uniqueness of solutions for small-energy weak immersions.

Framework allows analysis of energy quantization and finite-time singularities.

Provides a foundation for future studies on the global behavior of Willmore flows.

Abstract

We introduce a parametric framework for the study of Willmore gradient flows which enables to consider a general class of weak, energy-level solutions and opens the possibility to study energy quantization and finite-time singularities. We restrict in this first work to a small-energy regime and prove that, for small-energy weak immersions, the Cauchy problem in this class admits a unique solution.