On the stability of Killing cylinders in hyperbolic space

TL;DR

This paper investigates the stability of Killing cylinders in hyperbolic 3-space, analyzing their Morse index, stability criteria, and bifurcation into Delaunay surfaces across various support surfaces.

Contribution

It provides explicit Morse index calculations for Killing cylinders in hyperbolic space and explores their stability and bifurcation properties with respect to different support surfaces.

Findings

Explicit Morse index formulas for Killing cylinders.

Stability criteria analogous to Plateau-Rayleigh instability.

Delaunay surfaces bifurcate from Killing cylinders.

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.