Stationary surfaces for the moment of inertia with constant Gauss curvature

Abstract

Consider the energy $E_{\alpha }[\Sigma ]={\int }_{\Sigma }|p{|}^{\alpha }d\Sigma$, where $\Sigma$ 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.