2
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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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  • $\begingroup$ what is $\sigma$? $\endgroup$ – user99914 Apr 17 '17 at 10:15
  • $\begingroup$ By convexity you have $|A|^2 \leq H^2$. $\endgroup$ – Paul Bryan Apr 17 '17 at 11:29
  • $\begingroup$ Equation 3 is the Peter-Paul inequality (Young's inequality with $\eta$ wikipedia.org/wiki/Young%27s_inequality)? $\endgroup$ – Paul Bryan Apr 17 '17 at 11:33
  • $\begingroup$ @JohnMa I think $\sigma$ is $\delta$ in 2.1.Theorem. $\endgroup$ – lanse7pty Apr 17 '17 at 12:48
  • $\begingroup$ @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. $\endgroup$ – lanse7pty Apr 17 '17 at 12:55

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