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

Kuwert, Ernst; Schätzle, Reiner

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)

MSC:

53C44

Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Keywords:

Willmore surface; Willmore flow; Willmore functional; gradient flow; Willmore energy