I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken, specifically, I'm reading the following theorem:

$\textbf{Theorem 2.1}$ Let $R > 0$ and $\textbf{x}_0 \in R^{n+1}$ be arbitrary and define $\varphi(\textbf{x},t)= R^2 - |\textbf{x} - \textbf{x}_0| - 2nt$. If $\varphi_+$ denotes the positive part of $\varphi$ we have the estimate

$$v(\textbf{x},t) \varphi_+(\textbf{x},t) \leq \sup_{M_0} v \varphi_+$$

as long as $v(x,t)$ is defined everywhere on the support $\varphi_+$.

The authors denote $\textbf{x} := F(p,t)$ for a solution of the MCF, $\textbf{x}_0 := F(p,0) = M_0$ and define a gradient function $v := \left( \langle \nu, \omega \rangle \right)^{-1}$, where $\nu$ is the unit normal vector of the hypersurface and $\omega$ is some fixed vector such that $\langle \nu, \omega \rangle > 0$ and $|\omega| = 1$. They defined in the proof a function $\eta := \left( R^2 - r \right)^2$, where $r := |\textbf{x}| - 2nt$ (they assumed w.l.o.g. that $\textbf{x}_0 = 0$) and they got

$$\left( \frac{d}{dt} - \triangle \right) v^2 \eta \leq -12 v \nabla v \cdot \nabla \eta + \eta^{-1} \nabla \eta \cdot \nabla (v^2 \eta)$$

They stated

If we replace $\eta$ by $(\varphi_+)^2$ this computation remains valid on the support of $\varphi_+$ as long as $v$ is defined. The weak parabolic maximum principle then implies the result.

I would like to know what is this weak parabolic maximum principle.

This is what I thought about my question:

Initially, I thought that could be the weak parabolic maximum principle on $\mathbb{R}^n$ which is common to see in PDE courses, but the problem is that I'm working with a manifold which receive a local treatment, then I thought that the weak parabolic maximum principle on $\mathbb{R}^n$ could be extended to a manifold, but I couldn't extend the result.

I found a maximum principle applied on MCF in this lecture notes (it's the theorem $2.2.1$ on page $17$), but I couldn't see how this helps me to conclude the inequality of the theorem $2.1$ of the paper.

proofworks for the more general case. What you need, exactly, is for $\Omega$ to have compact closure in $M\times [0,\infty)$. If $M$ is compact and you're only interested in finite times, then you get this for free, of course. $\endgroup$