A Characterization of Minimal Rotational Surfaces in the de Sitter Space

In this paper, it is characterized the generating curves of minimal rotational surfaces in the de Sitter space S13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}_1^3$$\end{document} as solutions of a variational problem. More exactly, it is proved that these curves are the critical points of a potential energy functional involving the distance to a given plane among all curves of S12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}_1^2$$\end{document} with prescribed endpoints and fixed length. This extends the known Euler’s result that asserts that the catenary is the generating curve of the catenoid.