In this paper we consider the Willmore flow in three space dimensions. We prove that embedded surfaces can be driven to a self-intersection in finite time. This situation is in strict contrast to the behavior of hypersurfaces under the mean curvature flow, where the maximum principle prevents self-intersections, but very much analogous to the surface diffusion flow. The Willmore flow is a geometric evolution law in which the normal velocity of a moving surface equals the Laplace-Beltrami of the mean curvature plus some lower order terms. More precisely, we assume in the following that Γ_0 is a closed compact immersed and orientable surface in R~3.
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