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Stability of the free boundary Willmore problem

Anna Dall'Acqua, Fabian Rupp, Reiner Schätzle, Manuel Schlierf

TL;DR

The paper develops a new Łojasiewicz–Simon gradient inequality on analytic Banach manifolds and applies it to the Willmore problem with free boundary. By constructing generalized Gaussian coordinates and reformulating the Willmore energy and its boundary conditions in this framework, the authors establish a LS inequality for free boundary surfaces and prove stability and rigidity results. This yields global existence and convergence of the free boundary Willmore flow near local minimizers and a local rigidity statement near free boundary minimal surfaces, with quantitative stability estimates tied to the energy deficit. The approach provides a robust, diffeomorphism-invariant toolkit for higher-order free boundary variational problems and their gradient flows.

Abstract

We study the Willmore problem with free boundary by means of a new Łojasiewicz-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 Fréchet 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.

Stability of the free boundary Willmore problem

TL;DR

The paper develops a new Łojasiewicz–Simon gradient inequality on analytic Banach manifolds and applies it to the Willmore problem with free boundary. By constructing generalized Gaussian coordinates and reformulating the Willmore energy and its boundary conditions in this framework, the authors establish a LS inequality for free boundary surfaces and prove stability and rigidity results. This yields global existence and convergence of the free boundary Willmore flow near local minimizers and a local rigidity statement near free boundary minimal surfaces, with quantitative stability estimates tied to the energy deficit. The approach provides a robust, diffeomorphism-invariant toolkit for higher-order free boundary variational problems and their gradient flows.

Abstract

We study the Willmore problem with free boundary by means of a new Łojasiewicz-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 Fréchet 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.
Paper Structure (15 sections, 30 theorems, 196 equations)

This paper contains 15 sections, 30 theorems, 196 equations.

Table of Contents

  1. Introduction and main results
  2. The Łojasiewicz--Simon gradient inequality on Banach manifolds
  3. Willmore immersions and the Willmore flow with free boundary
  4. The Ł ojasiewicz-Simon inequality on Banach manifolds
  5. The Ł ojasiewicz--Simon inequality for the Willmore energy of free boundary surfaces
  6. Existence of generalized Gaussian coordinates
  7. Rewriting the free boundary problem in Gaussian coordinates
  8. The Ł ojasiewicz--Simon inequality for free boundary surfaces in Gaussian coordinates
  9. Stability of the free boundary Willmore flow
  10. Free boundary Willmore flow in Gaussian coordinates
  11. Stability of local minimizers under the flow
  12. Banach manifolds
  13. On analyticity of superposition operators
  14. Technical proofs
  15. Proof of \ref{['prop:reparam-as-fw']}

Key Result

Theorem 1

Let $V$, $Z$ be two Banach spaces, $\mathcal{M}$ an analytic $V$-Banach manifold, $u_0\in U_0\subset \mathcal{M}$ an open subset, and $\mathcal{E}\colon U_0\subset\mathcal{M}\to\mathbb{R}$, $\delta\mathcal{E}\colon U_0\subset\mathcal{M}\to Z$ be two analytic maps satisfying Then $\mathcal{E}$ satisfies a Ł ojasiewicz-Simon gradient inequality on $\mathcal{M}$ close to $u_0$, that is, there exist

Theorems & Definitions (70)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Corollary 1
  • proof
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • Remark 1
  • proof : Proof of \ref{['thm:loja-abstract-linear']}
  • ...and 60 more