Stationary surfaces for the moment of inertia with constant Gauss curvature

Abstract

Consider the energy Eα⁢[Σ]=∫Σ|p|α𝑑�Σ, where Σ is a surface in Euclidean space ℝ3 and α∈ℝ. We prove that planes and spheres are the only stationary surfaces for Eα with constant Gauss curvature. We also characterize these surfaces assuming that a principal curvature is constant or that the mean curvature is constant.