Willmore rigidity

Introduction

Willmore rigidity is a concept in differential geometry and mathematical physics that deals with the properties and behavior of surfaces that minimize the Willmore energy, which is the integral of the squared mean curvature. This topic has applications in various fields, including the study of minimal surfaces, string theory, and the analysis of fluid interfaces.

Key Insights

Constrained Willmore Surfaces:
- Surfaces can be characterized as minimizers of the Willmore energy subject to conformality constraints, which can be efficiently solved using gradient descent methods and Sobolev metrics.
Willmore Flow and Stability:
- The Willmore flow converges to a standard round sphere if the initial energy is small, with stability estimates for the barycenter, quadratic moment, enclosed volume, and averaged mean curvature .
Classical Dynamics and Willmore Functional:
- The dynamics of the rigid string in flat space-time can be described by the Euler-Lagrange equation for the Willmore functional, providing a useful reduction of the rigid string model.
Willmore Spheres in Riemannian Manifolds:
- Existence results for possibly branched Willmore spheres under various constraints (e.g., prescribed homotopy class, area) are established, with regularity for the minimizers.
Equivariant Willmore Surfaces:
- Classification of equivariant Willmore surfaces in homogeneous three spaces with rigidity four, with qualitative behavior encoded in parameters like base space curvature and bundle curvature.
Rigidity and Non-Rigidity Results:
- Quantitative rigidity results for minimizers of the Willmore functional among projective planes, with explicit counterexamples in codimension one.
Analysis of Willmore Surfaces:
- A new formulation for the Euler-Lagrange equation of the Willmore functional, solving the problem for immersions with L2-bounded second fundamental form and establishing regularity and compactness results.
Static Interface Profiles:
- Equilibrium conditions for fluid interfaces in contact with an elastic membrane, with the Willmore energy penalizing membrane bending and resulting in specific contact angle behaviors.

Conclusion

The study of Willmore rigidity encompasses various aspects, from the characterization and optimization of constrained Willmore surfaces to the analysis of their stability and dynamics. Key findings include the convergence of Willmore flow to round spheres under small initial energy, the classification of equivariant Willmore surfaces, and the establishment of rigidity and non-rigidity results. These insights contribute to a deeper understanding of the geometric and physical properties of surfaces governed by the Willmore functional.

Summary

Introduction

Key Insights

Conclusion

Sources and full results

Constrained willmore surfaces

This paper presents an efficient algorithm for constrained Willmore surfaces using triangle meshes, enabling the construction and manipulation of smooth surfaces with additional point, area, and volume constraints.

Asymptotic Estimates for the Willmore Flow With Small Energy

The Willmore flow converges to a standard round sphere with small initial energy, allowing for stability estimates for the barycenter, quadratic moment of the surface, and bounds on enclosed volume and averaged mean curvature.

Classical dynamics of the rigid string from the Willmore functional

The rigid string model in flat space-time can be reduced to the Willmore functional, allowing for the application of Nambu-Goto string dynamics results to Minkowski space-time.

Willmore spheres in compact Riemannian manifolds

This paper establishes the regularity for critical points of L 2 curvature functionals in compact Riemannian manifolds and proves the existence of possibly branched Willmore spheres under various constraints and curvature assumptions.

Equivariant Willmore surfaces in conformal homogeneous three spaces

Equivariant Willmore surfaces in conformal homogeneous three spaces with rigidity four can be classified by lifting elastic curves, with their qualitative behavior encoded in parameters (c, r) and the Euler Lagrange equation.

Asymptotic Estimates for the Willmore Flow With Small Energy

The Willmore flow converges to a standard round sphere with small initial energy, providing stability estimates for the barycenter, quadratic moment of the surface, and stability bounds for the enclosed volume and averaged mean curvature.

Rigidity and non-rigidity results for conformal immersions

This paper presents a quantitative rigidity result for conformal immersions of spheres with large Willmore energies, complementing recent rigidity results in [LaNg13].

Analysis aspects of Willmore surfaces

This paper presents a new formulation for the Euler-Lagrange equation of the Willmore functional for immersed surfaces in Rm, solving the longstanding open problem and establishing regularity for weak Willmore immersions with L2-bounded second fundamental form.

Static interface profiles for contact lines on an elastic membrane with the Willmore energy.

The Willmore energy balances surface tension in an elastic membrane, resulting in local flatness at the contact line and a transition layer near the contact line in the stiff limit.

The Willmore Flow with Small Initial Energy

A suitable blowup converges to a nonumbilic Willmore surface with small initial energy, resulting in long-time existence and convergence to a round sphere.

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