Assume $M$ is a n-dim convexity compact surface in $\mathbb R^{n+1}$, and $H$ is the mean curvature of $M$. How to show $$ \int_M \Delta H=0 $$ I get this question from the 43th page of Huisken, Gerhard, The volume preserving mean curvature flow, J. Reine Angew. Math. 382, 35-48 (1987). ZBL0621.53007.
As picture below, $\int \Delta H$ vanish. But seemly, there is not any condition making it is zero. So I guess for general convexity compact surface, we have $\int_M \Delta H=0$. But I fail to prove it. I just know to use the definition of mean curvature to calculate it.
It follows from the fact that $M$ is compact and without boundary and from the divergence theorem.