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?