# Two estimates in gradient estimate of mean curvature under volume preserving mean curvature flow

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?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ • what is $\sigma$? – user99914 Apr 17 '17 at 10:15
• By convexity you have $|A|^2 \leq H^2$. – Paul Bryan Apr 17 '17 at 11:29
• Equation 3 is the Peter-Paul inequality (Young's inequality with $\eta$ wikipedia.org/wiki/Young%27s_inequality)? – Paul Bryan Apr 17 '17 at 11:33
• @JohnMa I think $\sigma$ is $\delta$ in 2.1.Theorem. – lanse7pty Apr 17 '17 at 12:48
• @PaulBryan Thanks, do you mean $2N_1(1+h)|A|^2\le 2N_1(1+H_T)H_T^2 \le 2CN_1H_T^3$ ? $C$ is a constant. – lanse7pty Apr 17 '17 at 12:55