Catenaries and minimal surfaces of revolution in hyperbolic space

Abstract

We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight, where the weight is the distance with respect to a reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and also as a solution to a prescribed curvature problem involving certain vector fields. Finally, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane.Comment: 17 pages. Keywords: Catenary, extrinsic catenary, hyperbolic space, minimal surface, surface of revolutio