Let $H$ be the mean curvature of a hypersurface $\Sigma \subset M$ and $\phi \in C^\infty(\Sigma)$.
I want to calculate $\phi \langle \nabla H , \nabla \phi \rangle$.
Here I have two options:
$$\phi \langle \nabla H , \nabla \phi \rangle = 0 \tag{1}$$
and
$$\phi \langle \nabla H , \nabla \phi \rangle = - H \langle \nabla \phi , \nabla \phi \rangle = - H |\nabla \phi|^2 = H \phi \Delta \phi \tag{2}$$
since $H = const.$
Question:
Which one is correct?