This question is from a transformation of mean curvature flow. I want to find a precise initial conditions to keep $H \ge 1$ or $H\le 1$ under this flow.
Consider PDE on closed smooth manifold $M^n\subset \mathbb R^{n+1}$ $$ \partial_t H = \Delta H + |A|^2(H-1) $$ $H$ is mean curvature , $A$ is second fundamental form. We always assume $H \ge 0$. Obviously, if the $H\ge 1$ at every point when $t=0$. Then, it will be $H\ge 1$ for $t > 0$. For $ H\le 1$ , we have liking result.
But even for convex manifold, the initial condition , $H\ge 1$ or $H\le 1$ at everywhere, is too rough. Whether $\int _{M_0} (H-1) d\mu \ge 1$ can keep $H\ge 1$ for $t> T$? $M_0$ is the initial manifold, $T$ is a enough big number. In fact, I think it is not right. If so, what initial condition is suitable for keeping $H\ge 1$ or $H\le 1$ ?