# Distance function to mean curvature flow

In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the reader is referred to a paper of Evans and Spruck.

Claim. If $$(Q_t \mid t_0 \leq t \leq t_1)$$ is a smooth mean curvature flow in $$\mathbf{R}^n$$, then the signed distance function $$r$$ to $$Q_t$$ satisfies the inequality $$$$\tag{1} r\Big(\frac{\partial r}{\partial t} - \Delta r \Big) \geq 0$$$$ on the open neighborhood $$U$$ of $$(Q_t)$$ where $$r$$ is smooth.

Does the inequality $$(1)$$ really hold on the entirety of $$U$$?

Going through the Evans–Spruck paper, I could only make it work in a thin neighborhood of $$(Q_t)$$, say small enough that the spatial Hessian has $$\lvert r D^2 r \rvert < 1$$.

For the argument that follows in Ilmanen's book, it's perfectly fine to work with a thinner neighborhood of $$(Q_t)$$. What my question is getting at is whether it's necessary to update $$U$$ to a smaller open set in order for $$(1)$$ to hold.

This computation is only used when computing the evolution of the function $$\phi$$ defined on the previous page and $$\phi$$ is defined so it is supported in a small neighborhood of the smooth solution.
• Sure, but the size of the neighborhood isn't updated - the only thing imposed on it is that it be narrow enough that the Brakke flow and $Q$ do not initially intersect there, and that $r$ be smooth on it. Dec 15, 2022 at 8:10
• Why would it need to be updated? You are trying to get avoidance on a compact time interval and have a smooth compact flow so you just choose $\gamma$ small enough initially so the differential inequality is satisfied (I guess this should have been made more explicit and may be a gap/oversight in the proof but seems easy to fix). Dec 15, 2022 at 15:28
• I meant update in the sense that one might have to change $\gamma$ that was chosen on the previous page, 'so small that [...] $r$ is smooth on the set $U$', to a smaller value. I didn't mean that $\gamma$ would have to be updated along the flow. (Is that how you read my comment? That's how I interpreted your reply, anyway.) I didn't mean the question as a gotcha - obviously the size of $U$ plays no role in the argument, which is fine as is. I was just wondering whether I'd missed something, and the inequality really does hold wherever $r$ is smooth. Dec 15, 2022 at 15:58