The Willmore flow of tori of revolution

Abstract

We study long-time existence and asymptotic behavior for the L2-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 8π then the solution of the Willmore flow converges for t→∞ to the Clifford torus, possibly rescaled and translated. The energy threshold of 8π turns out to be optimal for such a convergence result. We give an application to the conformally constrained Willmore minimization problem.