Picture below is from 45th page of Huisken, Gerhard, The volume preserving mean curvature flow, J. Reine Angew. Math. 382, 35-48 (1987). ZBL0621.53007.
$N_2,N_1$ are constant. $M_0$ is initial manifold, and evolving under volume preserving mean curvature flow. $A$ is the second fundamental form. $H$ is mean curvature. $h(t)$ is the average of mean curvature $$ h(t)=\frac{\int_{M_t} H d\mu}{\int_{M_t} d\mu} $$ And $$ H_T=\max_{t\in[0,T]} \max_{M_t}H $$ Then, how to get the red line 2 and 3 ?
About getting the red line 2, I fail to deal $2N_1(1+h)|A|^2$. Because $C_6$ depends on $N_1$ and $M_0$. I feel part of $C_6$ is from $2N_1(1+h)|A|^2$.
About the red line 3, I am unfamiliar about the $\forall \eta>0$ .... Seemly, there is some all to know but I don't know, what is it?
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