Classification of ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia

TL;DR

This paper classifies ruled stationary surfaces in Lorentz-Minkowski space, showing that planes are the only cylindrical examples and providing explicit parametrizations based on the rulings' causal character.

Contribution

It offers the first complete classification of ruled stationary surfaces in Lorentz-Minkowski space, including explicit parametrizations and causal character analysis.

Findings

Planes are the only cylindrical stationary ruled surfaces.

Explicit parametrizations of all ruled stationary surfaces are provided.

Classification depends on the causal character of the rulings.

Abstract

In this paper, we study hypersurfaces in Lorentz-Minkowski space $\mathbb{L}^{n+1}$ that are stationary for the moment of inertia with respect to the origin. After giving examples and applications of the maximum principle, we classify, in $\mathbb{L}^3$, all stationary surfaces that are ruled surfaces. We prove that planes are the only cylindrical stationary surfaces. If the surface is not cylindrical, the classification depends on the causal character of the rulings. In addition, we provide explicit parametrizations of all ruled stationary surfaces.