Stability of cylinders in $\mathbb{E}(\kappa ,\tau )$ homogeneous spaces

We extend the classical Plateau-Rayleigh instability criterion in the $\mathbb{E}(\kappa ,\tau )$ spaces. We prove the existence of a positive number $L_0>0$ such that if a truncated circular cylinder of radius $\rho$ in $\mathbb{E}(\kappa ,\tau )$ has length $L>L_0$ then it is unstable. This number $L_0$ depends on $\kappa$, $\tau$ and $\rho$. The value $L_0$ is sharp under axially-symmetric variations of the surface. We also extend this result for the partitioning problem in $\mathbb{E}(\kappa ,\tau )$.