I'm trying to understand an argument that I've seen on two papers, which I do not fully understand but have a feeling for what is going on.
The claim is the following: Given that a domain $\Omega \subset \mathbb{R}^3$ is bounded, smooth and convex, if for a smooth function $c$ we have $\frac{\partial c}{\partial \nu} = 0$ on $\partial \Omega$, then $\frac{ \partial}{\partial \nu} |\nabla c|^2 \leq 0$ on $\partial \Omega$.
I've seen that the last term is equal to $- II(\nabla c)$ (1), and that for convex domains the second fundamental form is positive definite (on $\mathbb{R}^3$) (2) and now from $\frac{\partial c}{\partial \nu} = 0$ on $\partial \Omega$, we have that $\nabla c$ only has a tangential component on the boundary.
How can I formalize the fact that $\nabla c$ going alongside $\partial \Omega$ is carrying the second form information on sign?
Just as an observation, I don't know much geometry so I might have made some mistakes and would also appreciate any correction.
(1): Dal Passo, R., Garcke, H., Grün, G. (1998). On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal. 29:321–342.
