Removability of point singularities of Willmore surfaces. (English) Zbl 1078.53007

Kuwert, Ernst; Schätzle, Reiner

Ann. Math. (2) 160, No. 1, 315-357 (2004).

The Willmore functional is defined for immersions \(f\colon\Sigma \to \mathbb R^n\) of a closed surface \(\Sigma\) into Euclidean space. It is given by \[ W(f) = {1 \over 4} \int_\Sigma | H| ^2 \] where \(H\) denotes the mean curvature vector field. It is known that always \[ W(f) \geq 4\pi \] with equality only for round spheres. Moreover, if \(W(f) < 8\pi\), then \(f\) is an embedding. Critical points of the Willmore functional are called Willmore surfaces.
The authors show that certain point singularities of Willmore surfaces can be removed: The surface extends \(C^{1,\alpha}\) for all \(\alpha < 1\) into the point singularity (but not \(C^{1,1}\) as examples show). From this they deduce that the set of all closed surfaces \(\Sigma \subset \mathbb R^3\) of genus \(1\) and \(W(\Sigma) \leq 8\pi - \delta\), \(\delta>0\) fixed, is compact up to Möbius transformations under smooth convergence of compactly contained surfaces in \(\mathbb R^3\). There are also applications to long time existence of the corresponding Willmore flow.

Reviewer: Christian Bär (Potsdam)

MSC:

53A05	Surfaces in Euclidean and related spaces
49Q10	Optimization of shapes other than minimal surfaces

Keywords:

Willmore surface; Willmore flow; point singularity