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The Willmore flow with small initial energy. (English) Zbl 1035.53092
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.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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