  In this paper, we investigate the critical points of solutions to the
prescribed constant mean curvature equation with Neumann and Robin boundary
conditions respectively in a bounded smooth convex domain $\Omega$ of
$\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of
the critical points of solutions in a planar domain by using the local Chen &
Huang's comparison technique and the geometric properties of approximate
surfaces at the non-degenerate critical points. Secondly, we deduce the
uniqueness and non-degeneracy of the critical points of solutions in a
rotationally symmetric domain of $\mathbb{R}^{n}(n\geq3)$ by the projection of
higher dimensional space onto two dimensional plane.
