# Integration of Laplacian of mean curvature on manifold.

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. ## 1 Answer

It follows from the fact that $M$ is compact and without boundary and from the divergence theorem.

• I am so stuped. – lanse7pty Apr 10 '17 at 14:16
• ahah don't worry! :D It happens very often to me too! Are you also studying the mean curvature flow? – Onil90 Apr 10 '17 at 14:17
• Yes, but I just be a rookie. Only Huisken's The volume preserving mean curvature flow I almost finish. Thanks. – lanse7pty Apr 10 '17 at 14:20