Rotational Weingarten surfaces in Lorentz-Minkowski space

TL;DR

This paper introduces a new method based on geometric linear momentum to analyze and classify rotational Weingarten surfaces in Lorentz-Minkowski space, unifying different causal types of axes and providing explicit classifications.

Contribution

It develops a unified framework using geometric linear momentum to study rotational Weingarten surfaces, including classification of key families and characterization of quadric surfaces.

Findings

Reduced Weingarten conditions to first-order ODEs for momentum

Classified important families of rotational Weingarten surfaces

Characterized non-degenerate quadric surfaces via cubic Weingarten relations

Abstract

We propose a new approach to the study of rotational surfaces in Lorentz-Minkowski space based on the notion of the geometric linear momentum of the generatrix curves with respect to the axes of revolution. This technique allows us to reduce any Weingarten condition on the surface to a first-order ordinary differential equation for the momentum as a function of the distance to the corresponding axis, providing a unified framework that encompasses the three causal types of rotation axes. As a direct application, we classify important families of rotational Weingarten surfaces in this setting, including some linear and quadratic cases. Furthermore, we introduce the non-degenerate quadric surfaces of revolution in Lorentz-Minkowski space and characterize them in terms of a specific cubic Weingarten relation.