The mean curvature of a real algebraic surface $S$ in $\mathbb{R}^3$ defined as the zeroes of a polynomial $P$ of degree $d$ with real coefficients in three variables is given by the formula \begin{equation} H = \frac{1}{n |\nabla P|^3} \nabla P^t B \nabla P \end{equation} where $B = \Delta P \cdot I - Hess(P)$ (see for example Minimal hypersurfaces foliated by spheres, W. C. Jagy). If the surface is non-singular, then the locus $L$ where its mean curvature vanishes is the set of solutions of $ P = \nabla P^t B \nabla P = 0$. As $\nabla P^t B \nabla P$ is a polynomial of degree $3d-4$, classical results in topology of real algebraic varieties give some bounds on the number of connected components of the curve $L$.
As usual in topology of real algebraic varieties, we are trying to derive restrictions on the topology of $S$ in function of its degree $d$. Does the number of times the mean curvature of $S$ vanishes give any restriction on the topology of $S$ ?
