Stationary surfaces for the moment of inertia with constant Gauss curvature

Consider the energy $E_{\alpha }[\mathrm{\Sigma }]={\int }_{\mathrm{\Sigma }}\mathrm{\mid }p{\mathrm{\mid }}^{\alpha }d\mathrm{\Sigma }$, where $\mathrm{\Sigma }$ is a surface in Euclidean space \r^3 and α\in\r. We prove that planes and spheres are the only stationary surfaces for $E_{\alpha }$ with constant Gauss curvature. We also characterize these surfaces assuming that a principal curvature is constant or that the mean curvature is constant.