The Willmore flow with small initial energy.
The authors consider the \(L^2\) gradient flow for the Willmore functional. In another paper, they have proved that the curvature concentrates if a singularity develops. Here they show that a suitable blowup converges to a nonumbilic (compact or noncompact) Willmore surface. Furthermore, an \(L^\infty\) estimate is derived for the tracefree part of the curvature of a Willmore surface, assuming that its \(L^2\) norm (the Willmore energy) is locally small. One consequence is that a properly immersed Willmore surface with restricted growth of the curvature at infinity and small total energy must be a plane or a sphere. Combining the results they obtain long time existence and convergence to a round sphere if the total energy is initially small.
scientific article; zbMATH DE number 1782684