Two classification results for stationary surfaces of the least moment of inertia

A surface in Euclidean space \r^3 is said to be an $\alpha$-stationary surface if it is a critical point of the energy ${\int }_{\mathrm{\Sigma }}\mathrm{\mid }p{\mathrm{\mid }}^{\alpha }$, where α\in\r. We prove that all ruled $\alpha$-stationary surfaces are vector planes (for all $\alpha$) and a type of elongated helicoids (for $\alpha =1$). The second result of classification asserts that if $\alpha ≠-2,-4$, any $\alpha$-stationary surface foliated by circles must be a rotational surface. If $\alpha =-4$, the surface is the inversion of a plane, a helicoid, a catenoid or an Riemann minimal example. If $\alpha =-2$, we find many non-spherical cyclic $(-2)$-stationary surfaces.