The Plateau-Rayleigh instability of translating $\lambda$-solitons

Given a unit vector $\textbf{v}\in\mathbb{R}^3$ and $\lambda\in\mathbb{R}$, a translating $\lambda$-soliton is a surface in $\mathbb{R}^3$ whose mean curvature $H$ satisfies $H=\langle N,\textbf{v}\rangle+\lambda,\ |\textbf{v}|=1$, where $N$ is the Gauss map of the surface. In this paper, we extend the phenomenon of instability of Plateau-Rayleigh for translating $\lambda$-solitons of cylindrical type, proving that long pieces of these surfaces are unstable. We will provide explicit bounds on the length of these surfaces. It will be also proved that if a translating $\lambda$-soliton is a graph, then it is a minimizer of the weighted area in a suitable class of surfaces with the same boundary and the same weighted volume.