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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?

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    $\begingroup$ My instinctive reaction is that this might not be true without some further qualification of their respective behaviour at infinity. My (naive) reason for believing this is the analogy with the heat equation, related to the ill-posedness without growth restrictions. There are two solutions $u_0,u_1$ which respectively have $u_0(0,\cdot) = 0$ and $u_1(0,\cdot) \equiv 1$ so that $u_0$ grows extremely rapidly at infinity and $u_1 \equiv 1$. They immediately violate the inequality $u_0 < u_1$ that is satisfied at time $t = 0$. $\endgroup$
    – Leo Moos
    Commented Feb 12, 2021 at 18:02
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    $\begingroup$ I don’t think this heuristic from heat equation works for mean curvature flow, because the pathological behaviour you mentioned can be ruled out by Ecker-Huisken’s estimate for graphical MCF. Also, by a theorem of Chou-Zhu, for any smooth complete initial curve that divides the plane into two regions with infinite area, we have a unique solution to curve shortening flow for all positive time. $\endgroup$
    – Adrian Chu
    Commented Feb 12, 2021 at 20:29

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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

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

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  • $\begingroup$ I wonder if assuming "properly embeddedness" would help. $\endgroup$
    – Adrian Chu
    Commented Feb 13, 2021 at 2:41
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    $\begingroup$ The example above is properly embedded. It might help to assume something like a uniform (small) $r>0$ so $B_r(p)\cap \Sigma_0$ is a single (flat) graph over the tangent plane for all $p\in \Sigma_0$. This is the natural assumption for pseudolocality. $\endgroup$
    – RBega2
    Commented Feb 13, 2021 at 11:14

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