Stability of the free boundary Willmore problem

TL;DR

This paper investigates the stability and convergence properties of free boundary Willmore surfaces using a novel gradient inequality, providing new insights into their behavior near minimal surfaces.

Contribution

It introduces a new Lojasiewicz-Simon inequality for free boundary Willmore problems, enabling analysis without gradient-like Fréchet derivative representations.

Findings

Solutions near local minimizers exist globally and converge over time.

Quantitative stability of free boundary Willmore immersions is established.

A local rigidity result around free boundary minimal surfaces is proved.

Abstract

We study the Willmore problem with free boundary by means of a new {\L}ojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds. In contrast to previous works, we do not rely on a gradient-like representation of the Fr\'echet derivative, but merely on an inequality. For the free boundary Willmore flow, we prove that solutions starting sufficiently close to a local minimizer exist for all times and converge. In the static setting, we prove quantitative stability of free boundary Willmore immersions and a local rigidity result in a neighborhood of free boundary minimal surfaces.