Axisymmetric stationary surfaces for the moment of inertia

TL;DR

This paper classifies axisymmetric stationary surfaces in Euclidean space for a specific energy functional, revealing their geometric properties and characterizing special cases like closed and boundary surfaces.

Contribution

It provides a phase plane classification of axisymmetric stationary surfaces and proves that helicoidal stationary surfaces are necessarily rotational.

Findings

Classification of surfaces intersecting the axis orthogonally

Characterization of closed and boundary stationary surfaces for α=-2

Proof that helicoidal stationary surfaces are rotational

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.