Parabolic Maximum Principle on Geometric Flows I am trying to understand the main aspects of geometric flows. The parabolic maximum principle (PMP) plays an important role to bound geometric quantities, however, I haven't found a reference in which they show (for beginners) how to apply it and they only mention that it is an application of the PMP. To be precise, in Huisken and Ilmanen's paper "The inverse mean curvature flow and the Riemannian Penrose inequality" (https://projecteuclid.org/euclid.jdg/1090349447) they show that the evolution of the mean curvature is given by (equation 1.3):
$$\dfrac{\partial}{\partial t}H=\dfrac{1}{H^2}\Delta H-2\dfrac{|D H|^2}{H^3} -\dfrac{1}{H}\text{Ric}(\nu,\nu)-\dfrac{1}{H}|A|^2$$
and they mention the following:
"This equation is cause for optimism, because in view of the fact that $|A|^2 \geq \dfrac{H^2}{n−1}$, the parabolic maximum principle yields the curvature bound
$$\max\limits_{N_t} H^2 \leq \max_{N_0} H^2+C.$$
as long as the Ricci curvature is bounded below and the flow remains smooth."
It seems that the reason could be that from the evolution of H,
$$\dfrac{\partial}{\partial t}H \leq \dfrac{1}{H^2}\Delta H+\frac{|c|}{H}, $$
which calls for an application of the PMC, however from the usual statements of the PMC I have seen, I am not sure how to deal with the $1/H^2$. I also tried writing the evolution of $H^2$, but I got similar issues. To add up, the constant $C$ also puzzles me, since all the conclusions or uses I have seem of the PMP, point out to a bound of the type $\max\limits_{N_0} H^2$ (no constant).
 A: The $1/H^2$ is not the issue - since this is positive, the operator $L =  H^{-2}\Delta - \partial_t$ is parabolic, so the maximum principle applies. The reason you need a little subtlety (and that the $C$ appears in the final estimate) is that the Ricci bound is not necessarily positive. If it was, then we would have $LH \ge 0$ as you said, and so the maximum principle would immediately tell us that
$$\max\limits_{N_t} H^2 \leq \max_{N_0} H^2.$$ 
But what if our Ricci bound is negative, i.e. $\text{Rc}(\nu,\nu) \ge -c$ for some positive $c$? Then the best we can do is
$$\dfrac{\partial}{\partial t}H\le\dfrac{1}{H^2}\Delta H+\dfrac{c}{H}-\dfrac{H}{n-1},$$
so the problem is how to understand the effect of the "reaction term" $X(H) = c/H - H/(n-1)$. For small $H$ this can push $H$ to increase; but once $H^2 \ge c(n-1)$ we have $X(H)\le 0$ and thus $LH\ge 0$. If you follow through the proof of the maximum principle, this means that at a new maximum of $H^2$ we have $H^2 \le c(n-1)$, and so we get the bound
$$ \max_{N_t} H^2 \leq \max(\max_{N_0} H^2,c(n-1)),$$
so it looks like the authors are just using $C = c(n-1)$ in this case.
A more general and powerful way to handle reaction terms is to use an ODE comparison version of the maximum principle - if $X$ is Lipschitz, $\partial_t H \le \Delta H + X(H)$, $H(x,0) \le f(0)$ and $f'(t) =X(f(t))$, then $H(x,t) \le f(t)$ for all time. In this case you could solve the ODE $dH/dt = c/H - H/(n-1)$ to get a better, time-dependent bound on $H$.
I'm not aware of a nice self-contained introduction to maximum-principle methods. The best reference I can think of right now is chapter 6 of this book, which proves a few versions and has some examples of applications to Ricci flow.
