Let $F: M^n \to \mathbb R^{n+k}$ be an embedding and $F_t$ be a families of immersions so that $F_0=F$ and

$$\frac{\partial F_t}{\partial t} = \vec H$$

It is known that in hypersurface case ($k=1$), $F_t$ are all embedded as well, and this is not true in general. For example, one take two knotted disjoint circles in $\mathbb R^3$. After the mean curvature flow the two circle will touch each other at some time.

I am looking for an example such that $M$ is connected compact and $F_0$ is embedded, such that $F_t$ is not embedded at some $t$.

Also, are there any properties on $F_0$ and $M$ so that the mean curvature flow will stay embedded before singularity occurs?

  • $\begingroup$ Have you tried a trefoil knot? I am also pretty sure that you can connected your 2 circles "far out" such that you can get an example of the mean curvature flow which does not stay embedded (before singularities occur). $\endgroup$
    – Sebastian
    Oct 14 '14 at 7:21
  • $\begingroup$ @Sebastian: That should be a good candidate. Thanks! $\endgroup$ Oct 14 '14 at 7:43

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