Axisymmetric stationary surfaces for the moment of inertia

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