The avoidance principle for mean curvature flow says that, given two disjoint compact, embedded hypersurfaces, when we run the mean curvature flow, they will remain disjoint.


In fact it is known that we can relax the assumptions to “one is compact and embedded, the other one is just embedded”.


Can we relax the assumptions even more? e.g. to “two complete, embedded hypersurfaces”, or adding "properly embeddedness"? Or is there any known counterexample preventing us from doing so?



Avoidance principle of mean curvature flow for non-compact hypersurfaces?
You can't make such a relaxation, at least in the world of weak set flows. See Example 7.3 (specifically comment iv) of Ilmanen's paper [Generalized Flow of Sets by Mean Curvature on a Manifold](https://www.jstor.org/stable/24897049?seq=1#metadata_info_tab_contents)


I'd be surprised if restricting to classical flows helped, but am not entirely sure at this moment.


