The Willmore Flow of Tori of Revolution

Abstract

We study long-time existence and asymptotic behavior for the $L^2$-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. We show that if an initial datum has Willmore energy below $8\pi$ then the solution of the Willmore flow converges for $t\to \infty$ to the Clifford Torus, possibly rescaled and translated. The energy threshold of $8\pi$ turns out to be optimal for such a convergence result. We give an application to the conformally constrained Willmore minimization problem.

43 pages, To appear in Analysis & PDE

References in corpus (1)

• The elastic flow of curves in the hyperbolic plane

Cited by in corpus (6)

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• The free elastic flow for closed planar curves
• On the convergence of the Willmore flow with Dirichlet boundary conditions
• The Willmore flow of Hopf-tori in the $3$-sphere