Stability of cylinders in E(κ,τ) homogeneous spaces

We extend the classical Plateau-Rayleigh instability criterion in the E(κ,τ) spaces. We prove the existence of a positive number L₀>0 such that if a truncated circular cylinder of radius ρ in E(κ,τ) has length L>L₀ then it is unstable. This number L₀ depends on κ, τ and ρ. The value L₀ is sharp under axially-symmetric variations of the surface. We also extend this result for the partitioning problem in E(κ,τ).