The Willmore Flow of Tori of Revolution

TL;DR

This paper investigates the long-term behavior of the Willmore flow for tori of revolution, proving convergence to the Clifford Torus below an energy threshold of 8π, and explores related minimization problems.

Contribution

It establishes the optimal energy threshold for convergence of the Willmore flow of revolution tori to the Clifford Torus and applies this to conformally constrained minimization.

Findings

Convergence to Clifford Torus for energies below 8π.

Energy threshold of 8π is optimal for convergence.

Application to conformally constrained Willmore minimization.

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 $t \rightarrow \infty$ 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.