Catenaries and minimal surfaces of revolution in hyperbolic space

TL;DR

This paper introduces extrinsic catenaries in hyperbolic space, characterizes their properties, and links them to minimal surfaces of revolution, providing new insights into geometric analysis in hyperbolic geometry.

Contribution

It defines and characterizes extrinsic catenaries in hyperbolic space and connects them to minimal surfaces of revolution, offering novel geometric insights.

Findings

Extrinsic catenaries are critical points of a potential functional in hyperbolic space.

Generating curves of minimal surfaces of revolution are extrinsic catenaries.

An intrinsic characterization of some extrinsic catenaries is established.

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 as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, 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. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic.