A question in the book of Hamilton's Ricci flow I am reading the book "Hamilton's Ricci flow" by Chow, Lu and Ni. However, I am struggling to understand one step in P.172-173, which I think it is just a question in calculus and has nothing to do with the Ricci flow. Here is my question: It was deduced that (see (4.60) in P.172)
$$\tag{1}\frac{d}{dt}(A-C)
=(-2(A^2+AC+C^2)+B^2+AB+BC)(A-C).$$
Similarly, we have the equation for $\frac{d}{dt}(A-B)$ and $\frac{d}{dt}(B-C)$. 
Then it was claimed that if $A(0)\geq B(0)\geq C(0)$, then we have $A(t)\geq B(t)\geq C(t).$ This is the place I get lost. I just wonder how one can derive this from the evolution equation $(1)$, becasuse it seems to me that the coefficient of $A-C$ on the RHS of $(1)$, that is
$-2(A^2+AC+C^2)+B^2+AB+BC$, is not always positive. 
 A: You want to compare to a decaying exponential. On a time interval $[0,T]$ where $$G:=\left|-2(A^2+AC+C^2)+B^2+AB+BC\right| \le M_T,$$
we have the differential inequality $\partial_t (A-C) \ge -M_T(A-C)$ and thus $$A(t) -C(t) \ge (A(0)-C(0))e^{-M_T t}\ge0.$$
Since $G$ can be bounded in terms of $A,B,C$, we thus know that $A-C\ge0$ so long as $A(t),B(t),C(t)$ are finite. Perhaps also see one of my old questions, which is very similar.
A: The claim follows from existence and uniqueness of solutions to ODEs.  If the triple of functions $A(t)$, $B(t)$, $C(t)$ with $A(0) = B(0) = C(0)$  satisfy the system of ODEs, then we must have $A(t), B(t),$ and $C(t)$ constant for all time with $A(t) = B(t) = C(t) =A(0)$. This is on account of the constant functions $A(t) = B(t) = C(t) = A(0)$ also satisfying the system of ODEs.
Note:  I don't have the book in front of me, but this looks like the standard type of argument for the Ricci flow on three-dimensional Lie groups or homogenous spaces.  There is a paper by James Isenberg & Martin Jackson where a lot of this was originally done (https://projecteuclid.org/download/pdf_1/euclid.jdg/1214448265).  The paper is really well-written and quite accessible.
