Stationary surfaces for the moment of inertia with constant Gauss curvature

TL;DR

This paper characterizes the only stationary surfaces with constant Gauss curvature for a specific energy functional, showing that they are planes and spheres, and explores conditions involving principal and mean curvature.

Contribution

It provides a complete classification of stationary surfaces for the energy functional with constant Gauss curvature, including new characterizations under curvature constraints.

Findings

Planes and spheres are the only stationary surfaces for the energy with constant Gauss curvature.

Stationary surfaces are characterized when a principal curvature is constant.

Stationary surfaces are characterized when the mean curvature is constant.

Abstract

Consider the energy $E_\alpha[\Sigma]=\int_\Sigma |p|^\alpha\, d\Sigma$, where $\Sigma$ is a surface in Euclidean space $\r^3$ and $\alpha\in\r$. We prove that planes and spheres are the only stationary surfaces for $E_\alpha$ with constant Gauss curvature. We also characterize these surfaces assuming that a principal curvature is constant or that the mean curvature is constant.