Kuwert, Ernst; Schätzle, Reiner The Willmore flow with small initial energy. (English) Zbl 1035.53092 J. Differ. Geom. 57, No. 3, 409-441 (2001). 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. Reviewer: Huafei Sun (Beijing) Cited in 5 ReviewsCited in 58 Documents MSC: 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) Keywords:Willmore surface; Willmore flow; Willmore functional; gradient flow; Willmore energy PDF BibTeX XML Cite \textit{E. Kuwert} and \textit{R. Schätzle}, J. Differ. Geom. 57, No. 3, 409--441 (2001; Zbl 1035.53092) Full Text: DOI