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We introduce the concept of extrinsic catenary in the hyperbolic plane.Working in the hyperboloid model, we define an extrinsic catenary as the shapeof a curve hanging under its weight as seen from the ambient space. In otherwords, an extrinsic catenary is a critical point of the potential functional,where we calculate the potential with the extrinsic distance to a fixedreference plane in the ambient Lorentzian space. We then characterize extrinsiccatenaries in terms of their curvature and as a solution to a prescribedcurvature problem involving certain vector fields. In addition, we prove thatthe generating curve of any minimal surface of revolution in the hyperbolicspace is an extrinsic catenary with respect to an appropriate reference plane.Finally, we prove that one of the families of extrinsic catenaries admits anintrinsic characterization if we replace the extrinsic distance with theintrinsic length of horocycles orthogonal to a reference geodesic.

Catenaries and minimal surfaces of revolution in hyperbolic space