Catenaries and minimal surfaces of revolution in hyperbolic space

Luiz C. B. da Silva, Rafael López

Research output: Working paper/PreprintPreprint

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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.
Original languageEnglish
PublisherarXiv
Pages1-20
Number of pages20
Publication statusPublished - Nov 2022

Keywords

  • Catenary
  • Extrinsic catenary
  • Hyperbolic space
  • Minimal surface
  • Surface of revolution

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