Stability of cylinders in $\mathbb{E}(\hat{I}º,\ddot{I}„)$ homogeneous spaces

Abstract

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