I am reading Mean Curvature Flow and Isoperimetric Inequalities by Manuel Ritoré, Carlo Sinestrari, Vicente Miquel and Joan Porti and I am trying understand the use of an hypothesis. The Avoidance Principle is stated as follows:
$\textbf{Theorem 4.2}$ Let $M_0, N_0$ be two smooth closed surfaces and let $M_t, N_t$ be their evolutions under mean curvature flow. Let $T > 0$ be such that $M_t, N_t$ are both defined for all $t \in [0,T]$. Suppose that $M_0, N_0$ are disjoint. Then $M_t$ and $N_t$ are disjoint for all $t \in (0,T]$.
Felix Schulze in his lecture notes changes the hypothesis "$M_0, N_0$ be two smooth closed surfaces" by the hypothesis "at least $M_t$ or $N_t$ be compact" and gives the same proof that the theorem above using the strong maximum principle on page $10$ of his lecture notes (see page $14$ and $15$ for the proof of avoidance principle). This change in hypothesis led me to realize that I did not know how these hypothesis are used to prove the theorem, then I would like to know how these hypothesis are used in each case. I think that the difference is in how strong maximum principle apply in the proof. The strong maximum principle in Felix Schulze's notes is sufficient to prove the result (although I sis not understand this very well once that ask that the surface be closed seems sufficient for me). On the other hand, the authors in the book do not state the strong maximum principle, they only say that the result follows from the strong maximum principle, which led me to think that they use other version of the strong maximum principle
The only one maximum principle that I know is the theorem $2.2.1$ on page $17$ of this lecture notes. A more complete maximum principle states, under same hypothesis of the theorem $2.2.1$, "Moreover, if $M$ is connected and at some time $\tau \in (0,T')$ we have $u_{\text{max}} (\tau) = h(\tau)$, then $u = h$ in $M \times [0,\tau]$", which is the theorem $2.1.1$ of the book Lecture Notes on Mean Curvature Flow by Carlo Mantegazza. Although this is the strong maximum principle, I can not see how to apply it under the hypothesis of the theorem $4.2$ (supposing that this is the version of the strong maximum principle applied to the theorem $4.2$).
Thanks in advance!