Control of the isoperimetric deficit by the Willmore deficit

In the class of smoothly embedded surfaces of sphere type we prove that the isoperimetric deficit can be controlled by the Willmore deficit

The Willmore Flow of Tori of Revolution

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 $8\pi$ 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.Comment: 43 page