Solution of volume preserving mean curvature flow

Question

Consider the volume preserving mean curvature flow $$ \partial_tF(x,t)=(h(t)-H(x,t))\cdot\nu(x,t) ~~~x\in U, t\ge 0 \\ F(\cdot, 0 )=F_0 \\ h(t)=\frac{\int_{M_t}Hd\mu}{\int_{M_t}d\mu} $$ $F_0:R^n\supset U\rightarrow F_0(U)\subset M_0\subset R^{n+1}$ is local represent of $M_0$ . The relative notation can be saw in The volume preserving mean curvature flow. Let $A=\{h_{ij}\}$ is the second fundamental form. Then why $\forall m\ge 1$ $|\nabla^m A|^2$ is uniformly bounded $\Rightarrow$ the solution of volume preserving mean curvature flow exists for all time ?

Answer 1

The second fundamental form $A$ is (most of) the second derivative of $F$, and once you have bounds on all derivatives of $F$ the result follows by the Arzela-Ascoli theorem. Going from bounds on $\nabla^m A$ to bounds on $\partial^{m+2} F$ is a little involved - see e.g. Prop 2.4.9 of Mantegazza's Lecture Notes on Mean Curvature Flow.