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On the Stability of Killing Cylinders in Hyperbolic Space

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Abstract

In this paper we study the stability of a Killing cylinder in hyperbolic 3-space when regarded as a capillary surface for the partitioning problem. In contrast with the Euclidean case, we consider a variety of totally umbilical support surfaces, including horospheres, totally geodesic planes, equidistant surfaces and round spheres. In all of them, we explicitly compute the Morse index of the corresponding eigenvalue problem for the Jacobi operator. We also address the stability of compact pieces of Killing cylinders with Dirichlet boundary conditions when the boundary is formed by two fixed circles, exhibiting an analogous to the Plateau–Rayleigh instability criterion for Killing cylinders in the Euclidean space. Finally, we prove that the Delaunay surfaces can be obtained by bifurcating Killing cylinders supported on geodesic planes.

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Acknowledgements

Antonio Bueno has been partially supported by CARM, Programa Regional de Fomento de la Investigación, Fundación Séneca-Agencia de Ciencia y Tecnología Región de Murcia, reference 21937/PI/22. Rafael López is a member of the IMAG and of the Research Group “Problemas variacionales en geometría”, Junta de Andalucía (FQM 325). This research has been partially supported by MINECO/MICINN/FEDER grant no. PID2020-117868GB-I00, and by the “María de Maeztu” Excellence Unit IMAG, reference CEX2020-001105- M, funded by MCINN/AEI/10.13039/501100011033/ CEX2020-001105-M.

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  1. Departamento de Ciencias, Centro Universitario de la Defensa de San Javier, 30729, Santiago de la Ribera, Spain

    Antonio Bueno

  2. Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain

    Rafael López

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Bueno, A., López, R. On the Stability of Killing Cylinders in Hyperbolic Space. J Geom Anal 34, 281 (2024). https://doi.org/10.1007/s12220-024-01720-8

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