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.

enter image description here


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

  • $\begingroup$ I am so stuped. $\endgroup$ – lanse7pty Apr 10 '17 at 14:16
  • $\begingroup$ ahah don't worry! :D It happens very often to me too! Are you also studying the mean curvature flow? $\endgroup$ – Onil90 Apr 10 '17 at 14:17
  • 1
    $\begingroup$ Yes, but I just be a rookie. Only Huisken's The volume preserving mean curvature flow I almost finish. Thanks. $\endgroup$ – lanse7pty Apr 10 '17 at 14:20

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