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Consider the energyEα⁢[Σ]=∫Σ|p|α⁢dΣE_{\alpha}[\Sigma]=\int_{\Sigma}\lvert p\rvert^{\alpha}\,d\Sigma, where Σ is a surface in Euclidean spaceR3\mathbb{R}^{3}andα∈R\alpha\in\mathbb{R}. We prove that planes and spheres are the only stationary surfaces forEαE_{\alpha}with constant Gauss curvature. We also characterize these surfaces assuming that a principal curvature is constant or that the mean curvature is constant.

Stationary surfaces for the moment of inertia with constant Gauss curvature