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

Luiz C. B. da Silva (Lead / Corresponding author)

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
Pages (from-to)317-339
Number of pages23
JournalTohoku Mathematical Journal
Volume73
Issue number3
Early online date20 Sept 2021
DOIs
Publication statusPublished - Sept 2021

Keywords

  • Lorentz number
  • Lorentz-Minkowski space
  • Mean curvature
  • Skew curvature
  • Surface of revolution

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