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

TL;DR

This paper generalizes the classical instability criterion for cylinders to the $ ext{E}( ext{κ}, ext{τ})$ spaces, establishing a sharp length threshold for instability depending on space parameters and extending to partitioning problems.

Contribution

It introduces a new instability criterion for cylinders in $ ext{E}( ext{κ}, ext{τ})$ spaces, including a sharp length threshold and applications to partitioning problems.

Findings

Existence of a critical length $L_0$ for instability in $ ext{E}( ext{κ}, ext{τ})$ spaces.

The critical length $L_0$ depends on $ ho$, $ ext{κ}$, and $ ext{τ}$.

Extension of instability results to partitioning problems.

Abstract

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 $\mathbb{E}(\kappa,\tau)$.