Surfaces of revolution with prescribed mean and skew curvatures in Lorentz-Minkowski space

TL;DR

This paper studies surfaces of revolution in Lorentz-Minkowski space with prescribed mean and skew curvatures, transforming complex nonlinear ODEs into linear forms to find explicit parameterizations.

Contribution

It introduces a novel approach to prescribe curvatures by converting nonlinear ODEs into linear ones using hypercomplex numbers, enabling explicit solutions for surfaces of revolution.

Findings

Derived linear ODEs for prescribed mean and skew curvatures.

Provided explicit parameterizations of generating curves.

Extended methods to surfaces with lightlike axes.

Abstract

In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew ($S$) and mean ($H$) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that $H$ and $S$ can be interpreted in terms of the expected value and standard deviation of the normal curvature seen as a random variable, we address the problem of prescribed curvatures for surfaces of revolution. For surfaces with a non-lightlike axis and prescribed $H$, the strategy consists in rewriting the equation for $H$, which is initially a nonlinear second order Ordinary Differential Equation (ODE), as a linear first order ODE with coefficients in a certain ring of hypercomplex numbers along the generating curves: complex numbers for curves on a spacelike plane and Lorentz numbers for curves on a timelike plane. We also solve the problem for surfaces of revolution with a lightlike axis by using a certain ODE with real coefficients. On the other hand, for the skew curvature problem, we rewrite the equation for $S$, which is initially a nonlinear second order ODE, as a linear first order ODE with real coefficients. In all the problems, we are able to find the parameterization for the generating curves in terms of certain integrals of $H$ and $S$.