Grayson’s example

Abstract

We have seen in Chapter 7 that, if ∂E is smooth and compact, then for short times there exists a unique smooth compact mean curvature flow starting from ∂E. We have also seen in Example 3.21 that the sphere of radius R ₀ shrinks to a point in the finite time R ₀ ²/(2(n — 1)). This time can be interpreted as a singularity time of the flow, even if the evolving shere reduces to a point. In this chapter we describe an example, due to Grayson [159], of a smooth compact mean curvature flow in ℝ³ which develops a singularity, and does not reduce to a point at the same time: this example and the proof that we present here follows that of [159]. In [126] the reader can find various differences in the proof.