I have a question when reading Bennett Chow's book "The Ricci Flow: An Introduction."
The lemma 6.67 (in the page 219) in the book says that by the equation (6.6) (in the page 176), which asserts along the Ricci flow on a compact manifold, $$\frac{\partial}{\partial t}R=\Delta R+2|Ric|^2$$ where $R$ and $Ric$ are scalar and Ricci curvature at time $t,$ we have $$\frac{\partial}{\partial t}R\ge \Delta R$$ along the "normalized" Ricci flow. This seems nonsense to me since at the previous page (218) there claims that along the normalized Ricci flow we have $$\frac{\partial}{\partial t}R=\Delta R+2|Ric|^2-rR$$ where $r$ is the average of $R$ on the manifold at time $t,$ and I don't understand why we can derive $$2|Ric|^2-rR\ge 0.$$
In Hamilton's original paper "Three-manifolds with positive Ricci curvature," he treated this by Klingenber's theorem, so I got surprised when seeing that Chow could solve this just by maximum principle.
Any advice is welcome and thanks in advance!