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

Abstract

A surface in Euclidean space R3 is said to be an α-stationary surface if it is a critical point of the energy ∫Σ|p|α, where α∈R. These surfaces are characterized by the Euler-Lagrange equation H(p)=α⟨N(p),p⟩|p|2, p∈Σ, where H and N are the mean curvature and the normal vector of Σ. If α≠0, we prove that vector planes are the only ruled α-stationary surfaces. The second result of classification asserts that if α≠−2,−4, any α-stationary surface foliated by circles must be a surface of revolution. If α=−4, the surface is the inversion of a plane, a helicoid, a catenoid or an Riemann minimal example. If α=−2, and besides spheres centered at 0, we find non-spherical cyclic −-stationary surfaces.