Application of Sobolev inequality Sobolev inequality: For all Lipschitz functions $v$ on $M$ we have 
$$
\left( \int|v|^\frac{n}{n-1}\right)^{n-1/n} \le c(n)\left(\int _M |\nabla v| + \int _M H|v|\right)
$$
where $H$ is mean curvature of $M$.
Now, if we have
$$
\int_M |\nabla^m A|^p \le C(m,p)  ~~~~\forall m,p \ge 0
$$
Then how to show 
$$
|\nabla^m A| \le C(m)  ~~~~~~~ \forall m\ge 0~~~~?
$$
This is the last step of proof of 8.3 Lemma of  Flow by mean curvature of convex surfaces into spheres
 A: I don't know how to use Sobolev inequality to prove it. But from the 4.1 Theorem of The volume preserving mean curvature flow, I found a way to prove it.
First we have 
$$
\max_{M_t} |A|^2  \le C   ~~~~~\forall ~t\in[0,T)
$$
from 7.1 Theorem of Flow by mean curvature of convex surfaces into spheres, we have 
$$
\partial_t |\nabla^m A|^2 = \Delta |\nabla^m A|^2-2|\nabla^{m+1} A|^2+\sum_{i+j+k=m} \nabla ^i A*\nabla ^j A*\nabla^kA*\nabla^m A  \tag 1
$$
Now we prove it by induction on $m$. Suppose $|\nabla^n A|^2$ is uniformly bounded by $C_n$ for all $n\le m$. From (1), we have 
$$
\partial_t |\nabla^{m+1} A|^2 
\le \Delta |\nabla^{m+1} A|^2
+C_{11}(|\nabla^{m+1} A|^2 +1)
$$
choose $N\ge 2C_{11}$ and let $f=|\nabla^{m+1} A|^2 + N|\nabla^{m} A|^2 $, then we have 
$$
\partial_t f \le \Delta f -N|\nabla^{m+1}A|^2 +C_{12}
$$
the $-N|\nabla^{m+1}A|^2$ is from $\partial_t (N|\nabla ^m A|^2)$, then we have 
$$
\partial_t f \le \Delta f -Nf +C_{13}
$$
By Maximum principle, we have 
$$
f\le \frac{C_{13}}{N}+ e^{-Nt}(\max_{M_0} f- \frac{C_{13}}{N})
$$
so, we have proved Lemma 8.3 of Flow by mean curvature of convex surfaces into spheres.
In fact, I really want to know how to use Sobolev inequality to get it.
After read Sobolev imbedding theorem, I think it is a direct conclusion of Theorem 7.10 of Elliptic Partial Differential Equations of Second Order.

Theorem7.10 :
  $$
W_0^{1.p} \subset \left\{ 
\begin{aligned}
&L^{np/(n-p)}(\Omega)     &~~~p<n\\
&C^0(\overline \Omega)    &~~~p>n
\end{aligned}\right.
$$
  Firthermore, there exists a constant $C=C(n,p)$ such that, for any $u\in W_0^{1.p}$
  $$
\begin{aligned}
&||u||_{L^{np/(n-p)}} ~\le C ||Du||_{L^p}    &~~~p<n   \\
&\sup_\Omega |u| \le C|\Omega|^{1/n-1/p}  ||Du||_{L^p}  &~~~p>n 
\end{aligned}
$$

Since
$$
\int_M |\nabla^m A|^p \le C(m,p)  ~~~~\forall m,p \ge 0
$$
and $M$ is compact, let $p>dim M$, we have
$$
\nabla^{m-1}A\in W^{1,p}_0
$$
then 
$$
\max_M|\nabla^{m-1}A| \le  C(m,p,M)
$$
