Let $H$ be the mean curvature of a hypersurface $\Sigma \subset M$.
I know that
$$\frac{dH(t)}{dt} = \Delta_\Sigma\phi+ \text{Ric}(\nu,\nu)\phi + |A_{\Sigma}|^2\phi$$
but how about $g^{ik}\nabla_k H$; how to calculate it?
If I write
$$g^{ik}\nabla_k H = 0$$
since $H = const.$, is it true or not?
Thank you.