Stability of the free boundary Willmore problem

2026/01/26 by Anna Dall'Acqua, Fabian Rupp, Reiner Schätzle +1 · 1 voice
Mathematics · #math.AP #math.DG

paper · pdf

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.

Citations

• Convergence of gradient flows on knotted curves
• Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane
• The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow
• Dimension reduction for Willmore flows of tori: fixed conformal class and analysis of singularities
• Global existence for the Willmore flow with boundary via Simon's Li-Yau inequality
• Improved long time existence for the Willmore flow of surfaces of revolution with Dirichlet data
• Sharp quantitative rigidity results for maps from S² to S² of general degree
• Parabolic and Elliptic Schauder Theory on Manifolds for a Fourth-Order Problem with a First- and a Third-Order Boundary Condition
• Stability of the ball under volume preserving fractional mean curvature flow
• On the area-preserving Willmore flow of small bubbles sliding on a domain's boundary
• Convergence of Sobolev gradient trajectories to elastica
• A rigidity estimate for maps from S² to S² via the harmonic map flow
• Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method
• Non-linear Stability of Double Bubbles under Surface Diffusion
• On the Lojasiewicz-Simon gradient inequality on submanifolds
• Free boundary minimal surfaces in the unit ball : recent advances and open questions
• The free-boundary Brakke flow
• Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
• Local solutions to a free boundary problem for the Willmore functional
• Slowly converging Yamabe flows
• Sharp eigenvalue bounds and minimal surfaces in the ball
• Branched Willmore Spheres
• Theorems on Regularity and Singularity of Energy Minimizing Maps
• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Discussions

• New preprint on "Stability of the free boundary Willmore problem" by A. Dall'Acqua, F. Rupp, R. Schätzle, and M. Schlierf https://arxiv.org/abs/2601.18388 [bsky, 0 points, 0 comments]

Related

• Local solutions to a free boundary problem for the Willmore functional
• Gradient Flow for the Willmore Functional in Riemannian Manifolds of bounded Geometry
• Global existence for the Willmore flow with boundary via Simon's Li-Yau inequality
• Finite time singularities for the locally constrained Willmore flow of surfaces
• Towards a constrained Willmore conjecture