Consider a closed surface $\Sigma$ which bounds a solid $\Omega$ in ${\mathbb R}^3$. Assume some electric charges, say totally $Q$, is distributed on $\Sigma$ and reaches an "equilibrium" state. In this situation the electric field inside $\Omega$ should be $0$.
What can we say about the density $\rho$ of the charge in this equilibrium distribution? Is there a simple rule that relates $\rho$ to the geometry /curvature of $\Sigma$ (e.g. something like, $\rho$ proportional to the pointwise norm of the second fundamental form - but this is just speculation)? For example, if $\Sigma$ is a round sphere, then it is well known that $\rho$ is constant.