Axisymmetric stationary surfaces for the moment of inertia

Abstract

We investigate axisymmetric surfaces in Euclidean space that are stationary for the energy $E_\alpha=\int_\Sigma |p|^\alpha\, d\Sigma$. By using a phase plane analysis, we classify these surfaces when they intersect orthogonally the rotation axis. We also give some applications of the maximum principle characterizing the closed stationary surfaces and the compact stationary surfaces with boundary a circle when $\alpha=-2$. Finally, we prove that helicoidal stationary surfaces must be rotational surfaces.